#6635530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1686 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)(17,18,23,36,21,26,22,27)(19,28,37,39,30,24,31,43)(20,32,35,25,33,41,34,40)(29,44,47,42,45,48,46,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,29)(21,30)(22,31)(23,37)(25,38)(26,28)(27,39)(32,48)(33,45)(34,46)(35,47)(36,43)(40,42)(41,44);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20,21,33)(18,25,26,40)(19,29,30,45)(22,35,23,34)(24,38,28,42)(27,41,36,32)(31,47,37,46)(39,44,43,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(18,26)(24,28)(25,40)(27,36)(32,41)(38,42)(39,43)(44,48);(2,6)(5,8)(18,26)(19,30)(25,40)(27,36)(29,45)(31,37)(32,41)(46,47)

Polycyclic presentation:

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

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, 6542726);