#7507700

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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:

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

Polycyclic presentation:

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

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, 7414896);