#6812043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#35570	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,d^2=f,e^2=fh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719239);