#6726399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 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)(11,14)(12,15)(13,16)(17,18,21,24)(19,27,31,23)(20,32,33,25)(22,26,35,38)(28,36,30,40)(29,37,44,41)(34,45,46,39)(42,48,43,47);(2,5)(3,7)(9,11,13,12)(10,14,16,15)(17,19,21,31)(18,23,24,27)(20,29,33,44)(22,28,35,30)(25,37,32,41)(26,40,38,36)(34,42,46,43)(39,48,45,47);(2,5)(6,8)(9,11,13,12)(10,15,16,14)(18,24)(19,28)(20,33)(22,35)(23,36)(27,40)(29,43)(30,31)(37,48)(39,45)(41,47)(42,44);(2,5)(6,8)(17,20,21,33)(18,25,24,32)(19,29,31,44)(22,34,35,46)(23,37,27,41)(26,39,38,45)(28,42,30,43)(36,47,40,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(19,30)(22,35)(23,36)(26,38)(27,40)(28,31)(29,43)(34,46)(37,47)(39,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6633595);