#7509885

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

1350 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)(3,7,8,6)(9,10,13,15)(11,14,12,16)(17,18,23,27)(19,24,31,41)(20,25,33,44)(21,34,22,26)(28,37,46,47)(29,38,45,48)(30,40,36,39)(32,43,35,42);(2,5)(3,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,38)(26,39)(27,41)(28,32)(33,45)(34,40)(35,46)(37,43)(42,47)(44,48);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,28)(21,32)(22,35)(23,33)(24,37)(26,42)(27,44)(29,36)(30,45)(31,46)(34,43)(38,39)(40,48)(41,47);(1,3,4,8)(2,6,5,7)(10,15)(14,16)(17,20)(18,25)(19,29)(21,35)(22,32)(23,33)(24,38)(26,43)(27,44)(28,36)(30,46)(31,45)(34,42)(37,39)(40,47)(41,48);(17,21,23,22)(18,26,27,34)(19,30,31,36)(20,32,33,35)(24,39,41,40)(25,42,44,43)(28,45,46,29)(37,38,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,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,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, 7417081);