#8612183

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

4502 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=gi,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, 8519379);