#8928660

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

4243 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)(9,10,15,18)(11,16,19,24)(12,22,20,14)(13,17,23,21)(26,28)(27,35)(30,40)(33,38);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,13)(14,21)(15,19)(17,22)(18,24)(20,23)(25,26,31,28)(27,36,35,32)(29,38,37,33)(30,34,40,39);(1,2)(3,5)(4,6)(7,8)(11,19)(12,20)(14,22)(16,24)(25,27,31,35)(26,32,28,36)(29,30,37,40)(33,34,38,39);(9,12)(10,14)(11,13)(15,20)(16,21)(17,24)(18,22)(19,23)(25,28,31,26)(27,36,35,32)(29,38,37,33)(30,39,40,34);(1,3)(2,5)(4,7)(6,8)(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,21)(22,24)(25,29,31,37)(26,33,28,38)(27,30,35,40)(32,34,36,39);(1,2)(3,5)(4,6)(7,8)(9,14)(10,12)(11,21)(13,16)(15,22)(17,19)(18,20)(23,24)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835856);