#8607495

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(25,26,32,36)(27,30,31,33)(28,35,29,34)(37,38,39,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,24)(18,22)(20,23)(25,27,32,31)(26,33,36,30)(28,37,29,39)(34,38,35,40);(2,6)(3,7)(13,22)(16,20)(18,21)(23,24)(25,28,32,29)(26,34,36,35)(27,37,31,39)(30,40,33,38);(3,7)(4,8)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,29,32,28)(26,35,36,34)(27,37,31,39)(30,40,33,38);(1,2)(3,8)(4,7)(5,6)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)(25,30,32,33)(26,27,36,31)(28,38,29,40)(34,39,35,37);(1,4,5,8)(2,7,6,3)(9,11)(10,17)(12,15)(13,18)(14,19)(16,24)(20,23)(21,22)(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=hi,c^2=hi,d^2=g,e^2=g,f^2=h,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, 8514691);