#7510001

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(3,6,4,8)(9,10,13,15)(11,14,12,16)(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);(1,2)(3,6)(4,8)(5,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,24,26,39)(20,32,34,46)(21,29,23,36)(25,42,31,48)(27,40,35,41)(28,37,47,33)(38,43,45,44);(1,3,5,4)(2,6,7,8)(9,12,13,11)(10,14,15,16)(17,20,22,34)(18,25,26,31)(19,28,30,47)(21,33,23,37)(24,38,39,45)(27,43,35,44)(29,46,36,32)(40,42,41,48);(1,4,5,3)(2,6,7,8)(10,15)(14,16)(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=g,c^2=fg,d^2=f,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, 7417197);