#7507695

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414891);