#7512341

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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)(9,10,12,14)(11,15,16,13)(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)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,23,25,27)(20,31,32,47)(22,35,34,30)(24,39,40,48)(26,43,42,38)(28,36,46,33)(37,44,45,41);(10,14)(13,15)(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);(2,5)(6,8)(11,16)(13,15)(19,29)(23,27)(28,46)(30,35)(31,47)(37,45)(38,43)(39,48);(1,3,4,7)(2,6,5,8)(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=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7419537);