#6630440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(9,10,12,14)(11,15,16,13)(17,18,23,29)(19,25,32,42)(20,34,21,26)(22,28,24,38)(27,36,44,37)(30,43,35,39)(31,41,33,47)(40,46,48,45);(2,5)(6,8)(17,19)(18,25)(20,35)(21,30)(22,31)(23,32)(24,33)(26,43)(27,40)(28,41)(29,42)(34,39)(36,46)(37,45)(38,47)(44,48);(9,11,12,16)(10,13,14,15)(17,20,23,21)(18,26,29,34)(19,30,32,35)(22,37,24,36)(25,39,42,43)(27,28,44,38)(31,46,33,45)(40,47,48,41);(17,21,23,20)(18,27,29,44)(19,30,32,35)(22,37,24,36)(25,40,42,48)(26,28,34,38)(31,45,33,46)(39,47,43,41);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,28,29,38)(19,31,32,33)(20,36,21,37)(25,41,42,47)(26,27,34,44)(30,45,35,46)(39,48,43,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537636);