#7512114

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

1518 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)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,37)(30,36)(31,39)(32,38)(33,40)(41,45)(42,47)(43,46)(44,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,23,25,36)(20,28,31,42)(21,27,33,29)(24,35,39,47)(26,34,40,37)(32,41,44,43)(38,45,48,46);(10,14)(13,15)(19,27)(21,33)(23,34)(26,40)(28,41)(29,30)(32,44)(35,45)(36,37)(38,48)(42,43)(46,47);(2,5)(6,8)(11,16)(13,15)(17,20,22,31)(18,24,25,39)(19,28,30,42)(21,32,33,44)(23,35,36,47)(26,38,40,48)(27,41,29,43)(34,45,37,46);(1,3,4,7)(2,6,5,8)(18,25)(20,31)(23,36)(26,40)(28,42)(32,44)(34,37)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7419310);