#7086249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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)(17,18,22,27,23,28,37,44)(19,29,32,40,31,46,34,24)(20,25,36,43,35,42,48,39)(21,26,30,45,38,41,33,47);(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,40)(20,33,35,30)(21,36,38,48)(22,34,37,32)(25,41,42,26)(27,46,44,29)(39,47,43,45);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20,23,35)(18,25,28,42)(19,30,31,33)(21,32,38,34)(22,36,37,48)(24,26,40,41)(27,43,44,39)(29,45,46,47);(11,16)(13,15)(17,21,37,33,23,38,22,30)(18,26,44,47,28,41,27,45)(19,20,34,48,31,35,32,36)(24,39,46,42,40,43,29,25);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,31)(20,35)(24,40)(25,42)(29,46)(32,34)(36,48)(39,43)

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,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6993445);