#8651456

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8558652);