#6856508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fhi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257437, #72257512, #72264053
maximal normal subgroup of	#72257437, #72257512, #72264053
maximal quotient of	#72264692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763704);