#6485756

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

1406 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)(19,28)(20,25)(21,34)(22,27)(23,30)(24,33)(26,37)(29,38)(31,43)(32,42)(35,40)(36,41)(39,45)(44,47)(46,48);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,23,27,40)(20,29,32,44)(21,35,37,30)(24,38,41,47)(25,39,42,48)(26,43,34,28)(33,46,36,45);(10,14)(15,16)(17,20,22,32)(18,24,27,41)(19,29,31,44)(21,36,37,33)(23,38,40,47)(25,26,42,34)(28,39,43,48)(30,46,35,45);(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,39)(24,34)(26,41)(27,42)(28,38)(30,45)(31,44)(35,46)(36,37)(40,48)(43,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(20,32)(24,41)(25,42)(29,44)(33,36)(38,47)(39,48)(45,46)

Polycyclic presentation:

2^9|b^2=i,c^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,d^c=di,e^c=ei,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, 6392952);