#6809756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,28)(23,26,30,34)(24,27,32,29)(31,35,36,33);(2,3)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,30)(22,26,28,34)(24,31,32,36)(27,35,29,33);(1,2,4,3)(22,27)(23,30)(24,32)(26,33)(28,29)(34,35);(6,11)(10,13)(12,19)(18,20);(1,3,4,2)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,28)(23,30)(27,29)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,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, 6716952);