#6484123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1299 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,22,25)(19,23,28,37)(20,30,33,47)(21,35,32,26)(24,39,41,34)(27,42,31,46)(29,45,43,38)(36,48,40,44);(2,6)(7,8)(11,12)(14,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,37)(26,38)(27,33)(30,46)(32,43)(34,48)(35,45)(36,41)(39,44)(42,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,20,32,39,22,33,21,34)(18,24,35,47,25,41,26,30)(19,27,43,48,28,31,29,44)(23,36,45,46,37,40,38,42);(9,12,13,11)(10,14,15,16)(19,28)(23,37)(27,31)(29,43)(36,40)(38,45)(42,46)(44,48);(1,4,5,3)(2,8,6,7)(10,15)(14,16)(18,25)(19,28)(20,33)(26,35)(29,43)(34,39)(36,40)(42,46)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6391319);