#6814414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,20,26,35)(16,28,30,21)(17,22,27,36)(19,24,33,40)(25,41,43,34)(29,38,31,37)(32,44,42,39);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,35)(16,29,30,31)(17,33,27,19)(21,37,28,38)(22,40,36,24)(25,42,43,32)(34,44,41,39);(1,2,3,4)(13,16,19,32,18,30,33,42)(14,21,24,39,23,28,40,44)(15,25,27,29,26,43,17,31)(20,34,36,37,35,41,22,38);(6,10)(9,11)(16,30)(21,28)(25,43)(29,31)(32,42)(34,41)(37,38)(39,44);(7,12)(9,11)(13,17,19,15,18,27,33,26)(14,22,24,20,23,36,40,35)(16,31,32,25,30,29,42,43)(21,38,39,34,28,37,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721610);