#6485672

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 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)(17,18,22,27)(19,28,31,41)(20,25,32,39)(21,24,35,26)(23,34,37,30)(29,42,43,48)(33,38,45,40)(36,46,47,44);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,23,27,37)(20,29,32,43)(21,34,35,30)(24,28,26,41)(25,36,39,47)(33,46,45,44)(38,42,40,48);(10,14)(15,16)(18,24)(21,35)(23,28)(25,38)(26,27)(30,34)(33,45)(36,42)(37,41)(39,40)(44,46)(47,48);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,38)(26,40)(27,39)(28,42)(30,44)(31,43)(34,46)(35,45)(37,47)(41,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(20,32)(25,39)(29,43)(33,45)(36,47)(38,40)(42,48)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=fgi,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, 6392868);