#9005191

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

4389 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)(3,4)(5,6)(7,8)(25,26,31,35)(27,32,30,40)(28,29,34,33)(36,38,37,39);(2,6)(4,7)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,27,31,30)(26,32,35,40)(28,37,34,36)(29,39,33,38);(1,2,5,6)(3,7,8,4)(10,12)(11,17)(13,21)(22,24)(25,28,31,34)(26,33,35,29)(27,36,30,37)(32,39,40,38);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,29)(27,37)(30,36)(31,34)(32,39)(33,35)(38,40);(1,4,5,7)(2,3,6,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,29)(26,34)(27,38)(28,35)(30,39)(31,33)(32,37)(36,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,27)(26,32,35,40)(28,36,34,37)(29,39,33,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8912387);