#6485675

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,30)(19,31,35,46)(20,36,21,27)(22,29,23,28)(25,39,42,34)(26,38,44,37)(32,40,33,43)(41,48,45,47);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,24,35)(18,25,30,42)(20,33,21,32)(22,34,23,39)(26,43,44,40)(27,45,36,41)(28,31,29,46)(37,48,38,47);(10,14)(15,16)(17,20,24,21)(18,26,30,44)(19,32,35,33)(22,37,23,38)(25,40,42,43)(27,28,36,29)(31,45,46,41)(34,47,39,48);(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,27,30,36)(19,33,35,32)(22,37,23,38)(25,41,42,45)(26,28,44,29)(31,43,46,40)(34,47,39,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,28,30,29)(19,34,35,39)(20,37,21,38)(25,31,42,46)(26,36,44,27)(32,47,33,48)(40,45,43,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,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=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6392871);