#7511147

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1686 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,d^2=fgi,e^2=i,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

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, 7418343);