#6630857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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)(11,15,16,13)(17,18)(19,29)(20,25)(21,24)(22,37)(23,28)(26,32)(27,30)(31,43)(33,38)(34,41)(35,47)(36,39)(40,45)(42,44)(46,48);(2,5)(6,8)(17,19,30,36,23,32,22,21)(18,24,37,26,28,39,27,29)(20,31,44,46,34,45,35,33)(25,38,47,40,41,48,42,43);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,33)(22,35)(23,34)(24,38)(26,40)(27,42)(28,41)(29,43)(30,44)(32,45)(36,46)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(17,21,22,32,23,36,30,19)(18,26,27,24,28,29,37,39)(20,33,35,45,34,46,44,31)(25,40,42,38,41,43,47,48);(20,34)(25,41)(31,45)(33,46)(35,44)(38,48)(40,43)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538053);