#72338482

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#47929
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#1434170
prime radical	#6442555	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,13)(4,7)(5,10)(6,11)(8,12)(9,14)(15,29)(16,26)(17,25)(18,23)(19,24)(20,21)(22,28)(27,31)(30,32);(2,4,12,11)(3,14,17,16)(10,20,28,18)(15,30,31,19);(1,3)(2,9,12,26)(4,13,11,25)(5,19)(6,14)(7,16)(8,17)(10,27,28,29)(15,23)(18,24,20,32)(21,31)(22,30);(1,4,6,2,8,11,7,12)(3,9,16,13,17,26,14,25)(5,20,23,10,22,18,21,28)(15,24,19,27,31,32,30,29);(1,5)(2,10)(3,15)(4,18)(6,21)(7,23)(8,22)(9,24)(11,20)(12,28)(13,29)(14,19)(16,30)(17,31)(25,27)(26,32)

Polycyclic presentation:

2^10|b^2=i,c^2=h,d^2=gij,f^2=j,g^2=j,b^a=bg,c^a=cf,c^b=cgi,d^a=dghij,d^b=dgj,d^c=dgh,e^b=ei,e^c=egij,e^d=ei,f^a=fj,f^d=fj,g^a=gj,g^d=gj,g^e=gj,h^a=hj,h^d=hj,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338502
maximal normal subgroup of	#72338502

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);