#9053875

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

4301 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)(6,8)(9,10,16,21)(11,22,13,17)(12,23,24,18)(14,19,20,15)(25,26,31,36)(27,37,39,32)(28,35,30,33)(29,34,38,40);(3,7)(6,8)(9,11,16,13)(10,17,21,22)(12,20,24,14)(15,23,19,18)(25,26)(27,37)(28,33)(29,40)(30,35)(31,36)(32,39)(34,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,24)(10,18,21,23)(11,20,13,14)(15,22,19,17)(25,27,31,39)(26,32,36,37)(28,38,30,29)(33,40,35,34);(9,13,16,11)(10,17,21,22)(12,20,24,14)(15,18,19,23)(25,28,31,30)(26,33,36,35)(27,38,39,29)(32,40,37,34);(1,3)(2,6)(4,7)(5,8)(9,14,16,20)(10,19,21,15)(11,12,13,24)(17,18,22,23)(25,29)(26,34)(27,28)(30,39)(31,38)(32,33)(35,37)(36,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,19)(10,20,21,14)(11,23,13,18)(12,17,24,22)(25,30,31,28)(26,35,36,33)(27,29,39,38)(32,34,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961071);