#6693879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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)(17,18,21,25)(19,27,31,23)(20,32,34,24)(22,26,36,42)(28,39,30,37)(29,38,33,40)(35,45,46,41)(43,47,44,48);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,33)(21,31)(22,28)(24,40)(25,27)(26,37)(29,34)(30,36)(32,38)(35,44)(39,42)(41,48)(43,46)(45,47);(9,12,13,11)(10,14,15,16)(19,28)(20,34)(22,36)(23,37)(24,32)(26,42)(27,39)(29,44)(30,31)(33,43)(38,48)(40,47);(2,5)(6,8)(17,20,21,34)(18,24,25,32)(19,29,31,33)(22,35,36,46)(23,38,27,40)(26,41,42,45)(28,43,30,44)(37,47,39,48);(1,3,4,7)(2,6,5,8)(19,30)(22,36)(23,39)(26,42)(27,37)(28,31)(29,44)(33,43)(35,46)(38,48)(40,47)(41,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6601075);