#6635572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1694 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542768);