#9053869

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961065);