#6714690

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,45)(29,38)(31,43)(32,48)(33,40)(34,41)(35,42)(36,44)(37,46)(39,47);(1,2)(3,6)(4,5)(7,8)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,29,35,31)(24,39,40,48)(26,43,42,38)(28,36,46,34)(37,44,45,41);(10,13)(14,16)(17,20,21,33)(18,24,25,40)(19,28,30,46)(22,36,35,34)(23,37,27,45)(26,44,42,41)(29,32,31,47)(38,48,43,39);(1,3,4,7)(2,6,5,8)(18,25)(20,33)(23,27)(26,42)(28,46)(32,47)(34,36)(38,43);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(22,35)(23,38)(24,40)(27,43)(28,32)(30,31)(34,36)(37,48)(39,45)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fi,e^2=g,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, 6621886);