#6868712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	#31026	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775908);