#7507782

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7414978);