#9034439

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8941635);