#6814390

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(13,14,19,24)(15,20,27,35)(16,21,26,34)(17,22,31,39)(18,23,28,36)(25,33,42,44)(29,37,30,38)(32,40,41,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,24,35)(16,28,26,18)(17,30,31,29)(21,36,34,23)(22,38,39,37)(25,41,42,32)(33,43,44,40);(1,2,3,4)(13,16,18,15,19,26,28,27)(14,21,23,20,24,34,36,35)(17,29,32,25,31,30,41,42)(22,37,40,33,39,38,43,44);(6,10)(9,11)(17,31)(22,39)(25,42)(29,30)(32,41)(33,44)(37,38)(40,43);(7,12)(9,11)(13,17,18,32,19,31,28,41)(14,22,23,40,24,39,36,43)(15,25,26,30,27,42,16,29)(20,33,34,38,35,44,21,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721586);