#72338540

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	#53832
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#3343287
prime radical	#7504666	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3389 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)(4,6,11,7)(5,10,23,28)(9,14)(15,29)(16,26)(17,18)(19,22,21,24)(20,25)(27,31)(30,32);(2,4,12,11)(3,14,17,16)(10,21,28,19)(15,30,31,20);(1,3,8,17)(2,9)(4,18)(5,20,23,30)(6,16,7,14)(10,27)(11,13)(12,26)(15,24,31,22)(19,32)(21,25)(28,29);(1,4,6,2,8,11,7,12)(3,9,14,18,17,26,16,13)(5,21,24,10,23,19,22,28)(15,25,30,29,31,32,20,27);(1,5)(2,10)(3,15)(4,19)(6,22)(7,24)(8,23)(9,25)(11,21)(12,28)(13,29)(14,20)(16,30)(17,31)(18,27)(26,32)

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=hi,d^2=ghi,f^2=j,g^2=j,b^a=bg,c^a=cf,c^b=cgj,d^a=dghj,d^b=dgj,d^c=dgi,e^b=ei,e^c=eghij,e^d=ei,g^a=gj,g^c=gj,g^d=gj,g^e=gj,h^c=hj,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338568
maximal normal subgroup of	#72338568

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.8*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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*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, 3, 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);