#6682870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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)(15,19)(16,21)(17,20)(18,22)(23,39)(24,33)(25,34)(26,32)(27,37)(28,35)(29,38)(30,36)(31,40)(41,44)(42,43);(5,7,8,12)(6,9,10,11)(13,15,17,24)(14,19,20,33)(16,27,28,42)(18,23,30,26)(21,37,35,43)(22,39,36,32)(25,31,41,29)(34,40,44,38);(1,2,3,4)(14,20)(15,23)(18,30)(19,32)(21,35)(24,26)(25,27)(29,31)(33,39)(34,43)(37,44)(41,42);(6,10)(9,11)(15,24)(16,28)(19,33)(21,35)(23,26)(29,31)(32,39)(38,40);(7,12)(9,11)(13,16,17,28)(14,21,20,35)(15,25,24,41)(18,31,30,29)(19,34,33,44)(22,40,36,38)(23,27,26,42)(32,43,39,37)

Polycyclic presentation:

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

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, 6590066);