#6630919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(17,18,24,30)(19,27,33,41)(20,26,36,45)(21,25,32,42)(22,29,23,28)(31,48,34,44)(35,43,39,40)(37,47,38,46);(2,5)(6,8)(17,19,22,21,24,33,23,32)(18,25,28,27,30,42,29,41)(20,34,37,35,36,31,38,39)(26,43,46,44,45,40,47,48);(10,13)(14,16)(17,20,24,36)(18,26,30,45)(19,31,33,34)(21,39,32,35)(22,37,23,38)(25,40,42,43)(27,48,41,44)(28,46,29,47);(1,3,4,7)(2,6,5,8)(17,21,22,33,24,32,23,19)(18,27,28,42,30,41,29,25)(20,35,37,31,36,39,38,34)(26,44,46,40,45,48,47,43);(9,11,12,15)(10,14,13,16)(20,36)(26,45)(31,34)(35,39)(37,38)(40,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,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^b=di,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538115);