#9712273

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

4283 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,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, 9619469);