#8602860

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

4689 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=h,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, 8510056);