#6458006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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

1391 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,24)(22,26)(23,37)(25,34)(27,36)(29,43)(30,39)(32,45)(33,46)(35,42)(38,47)(40,44)(41,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,22)(23,30)(25,40)(26,28)(27,39)(29,33)(31,43)(34,47)(35,36)(37,42)(38,41)(44,48)(45,46);(11,15)(13,16)(17,20,23,34,22,33,36,48)(18,25,27,31,26,41,37,46)(19,29,30,44,21,32,35,47)(24,38,39,45,28,40,42,43);(11,15)(13,16)(17,21,22,19)(18,24,26,28)(20,32,33,29)(23,35,36,30)(25,38,41,40)(27,39,37,42)(31,45,46,43)(34,47,48,44);(1,3,4,7)(2,6,5,8)(19,21)(20,33)(24,28)(25,41)(30,35)(31,46)(34,48)(39,42)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365202);