#9545585

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

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,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, 9452781);