#9498335

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

4351 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)(3,4,8,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(26,32)(27,29)(34,38)(35,37);(1,3)(2,4)(5,8)(6,7)(10,14)(11,13)(12,20)(23,24)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(1,4)(2,3)(5,6)(7,8)(9,12,15,20)(10,17,14,21)(11,19,13,22)(16,23,18,24)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(9,13,15,11)(10,18,14,16)(12,19,20,22)(17,23,21,24)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9405531);