#9717403

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

4160 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=ghi,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, 9624599);