#6630414

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

1738 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)(17,18,21,24)(19,27,29,23)(20,31,32,40)(22,35,34,26)(25,36,39,33)(28,43,44,48)(30,38,46,42)(37,45,41,47);(2,5)(6,8)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,37)(26,38)(31,48)(32,44)(33,45)(34,46)(35,42)(36,47)(39,41)(40,43);(9,11,12,16)(10,13,14,15)(18,24)(23,27)(25,39)(26,35)(31,40)(37,41)(38,42)(43,48);(17,20,21,32)(18,25,24,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(1,3,4,7)(2,6,5,8)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^b=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, 6537610);