#6630486

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,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, 6537682);