#9303031

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9210227);