#8933009

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

4460 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)(9,10,14,15)(11,13,17,23)(12,19,20,16)(18,24,21,22)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,3,4,7)(2,5,6,8)(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)(3,5)(4,6)(7,8)(11,17)(13,23)(18,21)(22,24)(26,30)(28,37)(31,33)(34,40);(9,11,14,17)(10,13,15,23)(12,21,20,18)(16,24,19,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,20)(10,16,15,19)(11,18,17,21)(13,22,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,15)(12,22)(14,23)(16,21)(18,19)(20,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,e^2=g,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, 8840205);