#7510659

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 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)(3,4)(5,7)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,25)(21,24)(22,26)(23,30)(28,36)(29,40)(31,42)(32,37)(33,38)(34,41)(35,39)(43,46)(44,48)(45,47);(1,3)(2,6)(4,7)(5,8)(17,19,21,30)(18,23,24,27)(20,32,33,29)(22,28,35,31)(25,40,38,37)(26,42,39,36)(34,44,45,43)(41,46,47,48);(10,13)(14,15)(18,24)(19,28)(22,35)(23,36)(25,38)(27,42)(29,43)(30,31)(32,44)(34,45)(37,46)(40,48);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,20,21,33)(18,25,24,38)(19,29,30,32)(22,34,35,45)(23,37,27,40)(26,41,39,47)(28,43,31,44)(36,46,42,48);(1,4,5,6)(2,3,7,8)(20,33)(25,38)(29,32)(34,45)(37,40)(41,47)(43,44)(46,48)

Polycyclic presentation:

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

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, 7417855);