#7512311

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(17,18,22,27)(19,29,32,24)(20,25,21,26)(23,28,37,45)(30,47,48,39)(31,43,35,40)(33,46,38,41)(34,42,36,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,31,21,35)(23,38,37,33)(25,40,26,43)(28,46,45,41)(30,36,48,34)(39,44,47,42);(10,14)(13,15)(17,20,22,21)(18,25,27,26)(19,30,32,48)(23,36,37,34)(24,39,29,47)(28,44,45,42)(31,33,35,38)(40,41,43,46);(2,5)(6,8)(11,16)(13,15)(17,21,22,20)(18,26,27,25)(19,31,32,35)(23,36,37,34)(24,40,29,43)(28,44,45,42)(30,33,48,38)(39,41,47,46);(1,3,4,7)(2,6,5,8)(20,21)(25,26)(30,48)(31,35)(34,36)(39,47)(40,43)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7419507);