#6672985

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	#42797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(13,14,18,16)(15,23,27,20)(17,28,30,21)(19,22,32,29)(24,26,36,33)(25,34,40,37)(31,42,43,35)(38,39,44,41);(6,9)(7,11)(13,15,18,27)(14,20,16,23)(17,25,30,40)(19,33,32,26)(21,34,28,37)(22,36,29,24)(31,39,43,41)(35,44,42,38);(2,4)(13,16,18,14)(15,24,27,36)(17,28,30,21)(19,22,32,29)(20,33,23,26)(25,38,40,44)(31,35,43,42)(34,39,37,41);(6,9)(10,12)(13,17)(14,21)(15,25)(16,28)(18,30)(19,31)(20,34)(22,35)(23,37)(24,38)(26,41)(27,40)(29,42)(32,43)(33,39)(36,44);(5,7,8,11)(6,10,9,12)(15,26)(17,30)(19,32)(20,24)(21,28)(22,29)(23,36)(25,39)(27,33)(34,44)(37,38)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6580181);