#6810668

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42650	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6717864);