#6579943

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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)(17,18,22,27)(19,29,31,24)(20,25,21,26)(23,28,37,45)(30,47,48,39)(32,46,38,40)(33,43,35,41)(34,42,36,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,35)(22,31)(23,38)(25,41)(26,43)(27,29)(28,46)(30,34)(32,37)(36,48)(39,42)(40,45)(44,47);(10,14)(13,16)(17,20,22,21)(18,25,27,26)(19,30,31,48)(23,36,37,34)(24,39,29,47)(28,44,45,42)(32,35,38,33)(40,43,46,41);(1,3,4,7)(2,6,5,8)(20,34)(21,36)(23,37)(25,42)(26,44)(28,45)(30,33)(32,38)(35,48)(39,41)(40,46)(43,47);(17,21,22,20)(18,26,27,25)(19,30,31,48)(23,34,37,36)(24,39,29,47)(28,42,45,44)(32,35,38,33)(40,43,46,41)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487139);