#6808919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#35607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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:

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

Polycyclic presentation:

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

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, 6716115);