#7512117

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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)(17,18,22,28)(19,26,31,24)(20,32,35,25)(21,36,23,27)(29,39,47,43)(30,40,37,41)(33,42,46,48)(34,44,38,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,28,26)(20,33,35,46)(21,30,23,37)(25,42,32,48)(27,40,36,41)(29,38,47,34)(39,44,43,45);(10,14)(13,15)(17,20,22,35)(18,25,28,32)(19,29,31,47)(21,34,23,38)(24,39,26,43)(27,44,36,45)(30,46,37,33)(40,42,41,48);(2,5)(6,8)(11,16)(13,15)(17,19)(18,26)(20,29)(21,37)(22,31)(23,30)(24,28)(25,43)(27,40)(32,39)(33,34)(35,47)(36,41)(38,46)(42,44)(45,48);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,28,36)(19,30,31,37)(20,34,35,38)(24,40,26,41)(25,44,32,45)(29,46,47,33)(39,42,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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,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, 7419313);