#7508662

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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,23)(21,32)(22,25)(24,30)(27,28)(29,31);(1,3)(2,6)(4,7)(5,8)(17,19,22,30)(18,24,25,26)(20,31,32,28)(21,29,23,27);(9,11,12,16)(10,13,14,15)(18,25)(19,27)(21,23)(24,28)(26,31)(29,30);(2,7)(3,8)(10,14)(13,15)(17,20)(18,21)(19,28)(22,32)(23,25)(24,27)(26,29)(30,31);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,20,25,32)(19,29,30,27)(24,31,26,28)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=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, 7415858);