#6697882

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,27)(19,28,31,24)(20,25,34,43)(21,35,23,26)(29,45,47,38)(30,39,36,40)(32,48,46,41)(33,44,37,42);(9,10,13,14)(11,15,12,16)(17,19,22,31)(18,24,27,28)(20,32,34,46)(21,30,23,36)(25,41,43,48)(26,39,35,40)(29,37,47,33)(38,42,45,44);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,42)(27,43)(28,45)(30,46)(31,47)(32,36)(35,44)(39,41)(40,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,42,43,44)(29,46,47,32)(38,41,45,48);(10,14)(15,16)(19,30)(21,23)(24,40)(26,35)(28,39)(29,46)(31,36)(32,47)(33,37)(38,48)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=gi,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, 6605078);