#6630791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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

1488 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)(17,18,22,26)(19,27,28,42)(20,32,34,24)(21,36,29,25)(23,37,38,31)(30,45,33,43)(35,41,44,47)(39,46,40,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,28)(23,38)(27,42)(30,33)(31,37)(39,40)(43,45)(46,48);(17,19,29,37,22,28,21,31)(18,23,36,42,26,38,25,27)(20,33,44,46,34,30,35,48)(24,40,47,43,32,39,41,45);(10,14)(13,16)(17,20)(18,24)(19,30)(21,35)(22,34)(23,39)(25,41)(26,32)(27,43)(28,33)(29,44)(31,46)(36,47)(37,48)(38,40)(42,45);(1,3,4,7)(2,6,5,8)(20,34)(24,32)(30,33)(35,44)(39,40)(41,47)(43,45)(46,48)

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^c=di,e^d=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, 6537987);