#6643269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=fh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6550465);