#6630764

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629009
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

1484 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)(17,18,22,25)(19,23,28,38)(20,30,33,47)(21,35,32,26)(24,39,41,34)(27,42,31,46)(29,45,43,37)(36,48,40,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,38)(26,37)(27,33)(30,46)(32,43)(34,48)(35,45)(36,41)(39,44)(42,47);(17,20,32,39,22,33,21,34)(18,24,35,47,25,41,26,30)(19,27,43,48,28,31,29,44)(23,36,45,46,38,40,37,42);(10,14)(13,16)(18,25)(19,28)(24,41)(26,35)(27,31)(29,43)(30,47)(44,48);(1,3,4,7)(2,6,5,8)(20,33)(24,41)(27,31)(30,47)(34,39)(36,40)(42,46)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,d^b=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, 6537960);