#6629630

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,37)(29,43)(30,39)(31,45)(34,40)(36,41)(38,44)(42,48)(46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,38)(26,39)(27,37)(32,43)(33,44)(34,42)(35,45)(36,46)(40,48)(41,47);(10,14)(13,16)(19,28)(24,37)(29,42)(30,31)(38,47)(39,45)(43,48)(44,46);(1,3,4,7)(2,6,5,8)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,40)(27,41)(28,42)(30,44)(31,46)(32,35)(37,47)(39,48)(43,45);(17,21,23,22)(18,26,27,35)(19,30,28,31)(20,33,34,36)(24,39,37,45)(25,40,41,32)(29,44,42,46)(38,48,47,43)

Polycyclic presentation:

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

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, 6536826);