#7505357

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi

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, 7412553);