#7509430

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,27)(19,24,31,39)(20,25,34,42)(21,26,22,28)(29,37,46,48)(30,38,35,43)(32,40,45,47)(33,41,36,44);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,32)(21,30)(22,35)(23,31)(25,40)(26,38)(27,39)(28,43)(29,33)(34,45)(36,46)(37,41)(42,47)(44,48);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,41)(27,42)(28,44)(30,45)(31,46)(32,35)(38,47)(39,48)(40,43);(2,7)(3,8)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,43)(25,41,42,44)(29,45,46,32)(37,47,48,40);(1,4,5,6)(2,3,7,8)(18,27)(24,39)(25,42)(26,28)(37,48)(38,43)(40,47)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=f,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, 7416626);