#6630699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,27)(22,37)(23,28)(25,34)(26,31)(29,43)(30,39)(33,42)(35,47)(36,40)(38,44)(41,48)(45,46);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,28)(27,40)(32,43)(33,48)(34,44)(35,45)(37,39)(41,42)(46,47);(1,3,5,4)(2,7,6,8)(17,20,22,34,23,35,21,33)(18,25,27,32,28,42,37,47)(19,29,30,44,31,45,36,48)(24,38,40,43,26,41,39,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,19)(18,26)(20,29)(21,36)(22,30)(23,31)(24,28)(25,41)(27,39)(32,46)(33,48)(34,44)(35,45)(37,40)(38,42)(43,47);(17,21,23,22)(18,27,28,37)(19,30,31,36)(20,33,35,34)(24,39,26,40)(25,32,42,47)(29,44,45,48)(38,46,41,43)

Polycyclic presentation:

2^9|b^2=hi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,e^d=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, 6537895);