#6485879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,26)(20,25)(21,27)(22,30)(23,29)(24,28)(31,41)(32,39)(33,40)(34,38)(35,44)(36,43)(37,42)(45,47)(46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,22,20,24,33,23,32)(18,25,29,26,28,39,30,40)(21,31,36,34,35,45,37,46)(27,38,42,41,44,48,43,47);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,22,33,24,32,23,19)(18,26,29,39,28,40,30,25)(21,34,36,45,35,46,37,31)(27,41,42,48,44,47,43,38);(9,11,13,12)(10,15,14,16)(17,21,24,35)(18,27,28,44)(19,31,33,45)(20,34,32,46)(22,36,23,37)(25,38,39,48)(26,41,40,47)(29,42,30,43);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(18,28)(21,35)(25,39)(26,40)(29,30)(31,45)(34,46)(36,37)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6393075);