#6722435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6629631);