#7086248

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	C₂×#449	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)(17,18,24,37,23,27,22,28)(19,29,21,25,32,42,31,40)(20,33,38,41,35,26,36,48)(30,44,34,47,45,43,46,39);(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,25,27,40)(20,30,35,45)(21,24,31,22)(26,39,33,47)(28,42,37,29)(34,38,46,36)(41,44,48,43);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,30)(21,34)(22,36)(23,35)(24,38)(25,39)(27,33)(28,41)(29,43)(31,46)(32,45)(37,48)(40,47)(42,44);(11,16)(13,15)(17,21,22,19,23,31,24,32)(18,25,28,29,27,40,37,42)(20,34,36,30,35,46,38,45)(26,39,41,43,33,47,48,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,27)(20,35)(25,40)(28,37)(29,42)(30,45)(34,46)(36,38)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=gi,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, 6993444);