#72255657

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#201687
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#6470231	C₂
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=j,d^2=efj,f^2=j,b^a=bj,c^a=chi,c^b=cg,d^a=dh,d^b=dg,d^c=de,e^b=ei,e^d=eh,f^a=fj,f^c=fh,f^d=fh,f^e=fj,g^a=gj,g^d=gi,g^f=gj,h^b=hj,h^d=hj,i^d=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310790
maximal normal subgroup of	#72310790
maximal quotient of	#72255667

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