#6719201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,35,43)(28,38,46,45)(29,39,36,40)(32,41,47,48)(34,44,37,42);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,41,31,48)(26,39,43,40)(28,37,46,34)(38,44,45,42);(10,15)(14,16)(17,20,21,33)(18,24,25,31)(19,28,30,46)(22,37,35,34)(23,38,27,45)(26,44,43,42)(29,47,36,32)(39,41,40,48);(1,3,4,7)(2,6,5,8)(18,25)(20,33)(23,27)(26,43)(28,46)(32,47)(34,37)(39,40);(2,5)(6,8)(9,12,13,11)(10,16,15,14)(18,25)(19,29)(22,35)(23,39)(24,31)(27,40)(28,47)(30,36)(32,46)(34,37)(38,41)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6626397);