#6485773

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=hi,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,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, 6392969);