#7505156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1543 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7412352);