#6629937

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1785 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)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,35)(22,28)(23,38)(24,29)(27,36)(30,39)(31,40)(32,41)(33,43)(34,45)(37,44)(42,46)(47,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,24,21)(18,25,29,35)(20,32,34,30)(22,36,23,31)(26,41,45,39)(27,38,40,28)(33,46,37,47)(42,44,48,43);(10,14)(13,16)(17,20)(18,26)(19,30)(21,32)(22,37)(23,33)(24,34)(25,39)(27,42)(28,44)(29,45)(31,47)(35,41)(36,46)(38,43)(40,48);(17,21,24,19)(18,27,29,40)(20,32,34,30)(22,36,23,31)(25,38,35,28)(26,42,45,48)(33,47,37,46)(39,43,41,44);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,28,29,38)(19,31,21,36)(20,33,34,37)(25,40,35,27)(26,43,45,44)(30,46,32,47)(39,42,41,48)

Polycyclic presentation:

2^9|b^2=gi,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^b=ei,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, 6537133);