#7506439

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1987 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7413635);