#8602117

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

4679 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=g,f^2=g,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, 8509313);