#6682857

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	#36020	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)(13,14,18,22)(15,19,26,34)(16,20,28,36)(17,21,30,38)(23,31,25,33)(24,32,27,35)(29,37,41,44)(39,42,40,43);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,27)(17,23)(18,26)(20,35)(21,31)(22,34)(24,28)(25,30)(29,40)(32,36)(33,38)(37,43)(39,41)(42,44);(1,2,3,4)(15,23)(17,30)(19,31)(21,38)(24,39)(25,26)(27,40)(29,41)(32,42)(33,34)(35,43)(37,44);(6,10)(9,11)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,37)(22,36)(23,39)(25,40)(26,27)(30,41)(31,42)(33,43)(34,35)(38,44);(7,12)(9,11)(15,23)(16,28)(17,30)(19,31)(20,36)(21,38)(24,40)(25,26)(27,39)(32,43)(33,34)(35,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,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, 6590053);