#7086505

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=g,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, 6993701);