#9243527

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

4387 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,19,21,15)(12,22,23,16)(13,17,20,24)(25,26)(27,30)(28,31)(29,33)(32,35)(34,38)(36,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,18,19)(12,13,23,20)(16,17,22,24)(25,27,29,35)(26,30,33,32)(28,34,37,40)(31,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,16,18,22)(11,20,21,13)(15,24,19,17)(25,26,29,33)(27,32,35,30)(28,36,37,31)(34,38,40,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,13)(14,23)(15,17)(18,22)(19,24)(20,21)(25,28,29,37)(26,31,33,36)(27,34,35,40)(30,38,32,39);(9,13)(10,17)(11,12)(14,20)(15,16)(18,24)(19,22)(21,23)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,37)(38,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,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, 9150723);