#6599124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(11,16)(12,14)(13,15)(17,18,22,26)(19,23,28,38)(20,30,32,42)(21,34,36,25)(24,35,41,33)(27,43,31,48)(29,46,45,39)(37,47,40,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,39)(26,38)(27,32)(30,48)(33,47)(34,46)(35,44)(36,45)(37,41)(42,43);(11,12)(14,16)(17,20)(18,24)(19,27)(21,35)(22,32)(23,37)(25,42)(26,41)(28,31)(29,47)(30,34)(33,36)(38,40)(39,48)(43,46)(44,45);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(20,32)(24,41)(27,31)(30,42)(33,35)(37,40)(43,48)(44,47);(11,12)(14,16)(19,28)(20,32)(23,38)(24,41)(29,45)(30,42)(33,35)(39,46)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506320);