#6697435

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1261 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)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,37)(29,45)(31,43)(32,48)(33,39)(34,42)(35,40)(36,44)(38,47)(41,46);(10,13)(11,15)(17,19)(18,23)(20,32)(21,30)(22,28)(24,27)(25,41)(26,43)(29,36)(31,35)(33,46)(34,47)(37,40)(38,44)(39,48)(42,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(20,33)(22,35)(23,37)(27,43)(29,32)(30,31)(38,48)(41,45)(42,44)(46,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,24)(20,33)(23,27)(26,40)(29,47)(32,46)(34,36)(37,43);(17,20,21,33)(18,25,24,39)(19,29,30,47)(22,36,35,34)(23,38,27,45)(26,44,40,42)(28,46,31,32)(37,41,43,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6604631);