#8933748

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

4449 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)(11,13)(12,19)(14,15)(16,20)(17,23)(18,24)(21,22)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,5)(6,7)(10,15)(11,17)(12,20)(22,24)(25,26,29,30)(27,31,35,33)(28,32,37,36)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(11,17)(13,23)(18,21)(22,24)(26,30)(28,37)(31,33)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,21)(14,17)(15,23)(16,24)(18,20)(19,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12)(10,16)(11,18)(13,22)(14,20)(15,19)(17,21)(23,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(9,13)(10,11)(12,22)(14,23)(15,17)(16,18)(19,21)(20,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840944);