#9729225

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

4345 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)(3,4)(5,8)(6,7)(9,10,14,15)(11,12,16,19)(13,21,23,17)(18,22,24,20)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(1,2)(3,6)(4,7)(5,8)(10,15)(12,19)(17,21)(20,22)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,2,5,8)(3,4,7,6)(10,15)(11,16)(17,21)(18,24)(26,31)(30,37)(32,35)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,16)(10,12,15,19)(13,18,23,24)(17,20,21,22)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,15)(13,22)(14,19)(17,18)(20,23)(21,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,20,19,22)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9636421);