#6635594

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1653 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,22,27,23,28,24,29)(19,30,37,40,21,25,32,43)(20,33,38,26,36,44,35,42)(31,45,46,41,34,48,47,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,25)(20,34)(21,23)(22,37)(24,32)(26,41)(27,43)(28,30)(29,40)(31,36)(33,45)(35,46)(38,47)(39,42)(44,48);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,34)(22,35)(23,36)(24,38)(25,39)(27,33)(28,42)(29,44)(30,41)(32,46)(37,47)(40,45)(43,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,21,23,19)(18,25,28,30)(20,34,36,31)(22,32,24,37)(26,39,42,41)(27,43,29,40)(33,48,44,45)(35,46,38,47);(2,6)(5,8)(17,22,23,24)(18,27,28,29)(19,32,21,37)(20,35,36,38)(25,40,30,43)(26,33,42,44)(31,46,34,47)(39,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6542790);