#6535184

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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)(10,13)(14,16)(17,18,23,27)(19,24,29,38)(20,32,34,40)(21,26,22,35)(25,36,41,33)(28,42,43,48)(30,39,31,45)(37,46,47,44);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,43)(35,45)(36,46)(40,48)(41,47);(10,13)(14,16)(17,20,23,34)(18,25,27,41)(19,28,29,43)(21,33,22,36)(24,37,38,47)(26,40,35,32)(30,44,31,46)(39,48,45,42);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,27,35)(19,30,29,31)(20,33,34,36)(24,39,38,45)(25,40,41,32)(28,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

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, 6442380);