#6630814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,31)(21,35)(22,27)(24,33)(25,29)(26,36)(28,42)(30,41)(32,45)(34,46)(37,43)(38,44)(39,47)(40,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,23,27,25)(20,32,34,28)(21,30,36,44)(24,39,40,37)(26,38,35,41)(31,45,46,42)(33,47,48,43);(17,20,21,33,22,34,36,48)(18,24,26,31,27,40,35,46)(19,28,30,43,29,32,44,47)(23,37,38,42,25,39,41,45);(10,14)(13,16)(17,19)(18,25)(20,32)(21,30)(22,29)(23,27)(24,37)(26,41)(28,34)(31,42)(33,47)(35,38)(36,44)(39,40)(43,48)(45,46);(1,3,4,7)(2,6,5,8)(19,29)(23,25)(28,32)(30,44)(37,39)(38,41)(42,45)(43,47)

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,d^b=di,e^b=ei,d^c=di,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, 6538010);