#6669112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,28)(17,21)(18,22)(19,27)(20,29)(24,34)(25,33)(26,32)(30,41)(31,36)(35,42)(37,39)(38,43)(40,44);(6,9)(7,11)(13,15,17,27)(14,19,21,23)(16,25,29,39)(18,24,31,26)(20,33,28,37)(22,32,36,34)(30,38,42,40)(35,43,41,44);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(18,31)(19,32)(22,36)(23,34)(25,38)(26,27)(30,42)(33,43)(35,41)(37,44)(39,40);(6,9)(10,12)(13,16)(14,20)(15,25)(17,29)(18,30)(19,33)(21,28)(22,35)(23,37)(24,38)(26,40)(27,39)(31,42)(32,43)(34,44)(36,41);(14,21)(15,26)(16,29)(18,31)(19,32)(23,34)(24,27)(25,38)(33,44)(35,41)(37,43)(39,40)

Polycyclic presentation:

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

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, 6576308);