#9729227

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

4371 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,f^2=gh,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, 9636423);