#6693854

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1299 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)(9,10,13,15)(11,14,12,16)(17,18,22,26)(19,23,30,38)(20,31,32,24)(21,25,34,40)(27,35,29,37)(28,42,44,36)(33,45,46,39)(41,48,43,47);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,38)(20,28,32,44)(21,27,34,29)(24,36,31,42)(25,35,40,37)(33,41,46,43)(39,47,45,48);(9,12,13,11)(10,14,15,16)(19,27)(20,32)(21,34)(23,35)(24,31)(25,40)(28,43)(29,30)(36,48)(37,38)(41,44)(42,47);(2,5)(6,8)(17,20,22,32)(18,24,26,31)(19,28,30,44)(21,33,34,46)(23,36,38,42)(25,39,40,45)(27,41,29,43)(35,47,37,48);(1,3,4,7)(2,6,5,8)(19,27)(21,34)(23,35)(25,40)(28,41)(29,30)(33,46)(36,47)(37,38)(39,45)(42,48)(43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6601050);