#7086506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

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, 6993702);