#6488871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(11,14)(12,15)(13,16)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,35)(28,41)(29,42)(31,46)(32,39)(33,48)(36,43)(37,45)(38,47)(40,44);(11,12)(14,15)(17,19,21,30,22,28,35,43)(18,23,26,27,25,36,34,41)(20,31,33,47,32,29,44,45)(24,38,40,46,39,37,48,42);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(19,28)(20,32)(23,36)(24,39)(27,41)(30,43)(33,44)(40,48);(9,12,13,11)(10,15,16,14)(17,20,22,32)(18,24,25,39)(19,29,28,31)(21,33,35,44)(23,37,36,38)(26,40,34,48)(27,42,41,46)(30,45,43,47);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(18,25)(20,32)(23,36)(26,34)(27,41)(29,31)(33,44)(45,47)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6396067);