#6630914

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(11,14)(12,13)(15,16)(17,18)(19,29)(20,25)(21,24)(22,39)(23,28)(26,32)(27,30)(31,43)(33,46)(34,42)(35,44)(36,48)(37,40)(38,41)(45,47);(2,5)(6,8)(17,19,30,38,23,32,22,21)(18,24,39,26,28,41,27,29)(20,33,47,37,35,31,36,34)(25,42,48,43,44,40,45,46);(10,13)(14,16)(17,20,23,35)(18,25,28,44)(19,31,32,33)(21,37,38,34)(22,36,30,47)(24,40,41,42)(26,46,29,43)(27,45,39,48);(1,3,4,7)(2,6,5,8)(17,21,22,32,23,38,30,19)(18,26,27,24,28,29,39,41)(20,34,36,31,35,37,47,33)(25,43,45,42,44,46,48,40);(9,11,12,15)(10,14,13,16)(20,35)(25,44)(31,33)(34,37)(36,47)(40,42)(43,46)(45,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci,d^c=di,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, 6538110);