#8974563

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

4347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=g,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, 8881759);