#72312654

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#6487485	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=fh,c^2=fi,e^2=fij,b^a=bj,c^a=cf,c^b=chij,d^c=dj,e^a=ef,e^b=efg,e^c=efj,e^d=efi,g^b=gi,g^c=gij,g^d=gj,g^e=gj,h^c=hj,h^e=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312676, #72312945
maximal normal subgroup of	#72312676, #72312945

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.6);;
gap> SetPower(coll, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*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, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);