#6754009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6661205);