#7512179

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,34)(22,26)(23,27)(28,36)(29,37)(30,38)(31,39)(32,41)(33,42)(35,40)(43,46)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,27,39)(20,29,33,45)(21,30,22,28)(25,37,42,48)(26,36,34,38)(32,43,35,44)(40,47,41,46);(10,14)(13,15)(19,28)(21,22)(24,36)(26,34)(29,43)(30,31)(32,35)(37,46)(38,39)(40,41)(44,45)(47,48);(2,5)(6,8)(11,16)(13,15)(17,20,23,33)(18,25,27,42)(19,29,31,45)(21,35,22,32)(24,37,39,48)(26,41,34,40)(28,43,30,44)(36,46,38,47);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,34)(19,30,31,28)(20,32,33,35)(24,36,39,38)(25,40,42,41)(29,43,45,44)(37,47,48,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7419375);