#9015281

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

4648 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=ghi,e^2=i,f^2=h,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, 8922477);