#72338481

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#59202
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#893870
prime radical	C₂×#3018	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3922 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=hi,d^2=gi,f^2=j,g^2=j,b^a=bg,c^a=cf,c^b=cgi,d^a=dgj,d^b=dgj,d^c=dghij,e^b=ei,e^c=egij,e^d=ei,g^a=gj,g^d=gj,g^e=gj,h^c=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, 1, F.8);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.9);;
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.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
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, 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, 3, 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);