#9713337

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

4661 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)(9,10,16,18)(11,19,13,17)(12,22,14,15)(20,21,23,24)(25,26)(27,35)(28,38)(29,32)(30,34)(31,33)(36,39)(37,40);(1,3,6,4)(2,7,5,8)(9,11,16,13)(10,17,18,19)(12,20,14,23)(15,21,22,24)(25,27,32,31)(26,33,29,35)(28,37,30,36)(34,40,38,39);(9,12,16,14)(10,15,18,22)(11,20,13,23)(17,21,19,24)(25,28,32,30)(26,34,29,38)(27,36,31,37)(33,39,35,40);(9,13,16,11)(10,17,18,19)(12,20,14,23)(15,24,22,21)(25,29,32,26)(27,35,31,33)(28,34,30,38)(36,39,37,40);(1,4,6,3)(2,7,5,8)(9,14,16,12)(10,15,18,22)(11,20,13,23)(17,24,19,21)(25,30,32,28)(26,34,29,38)(27,37,31,36)(33,39,35,40);(1,5,6,2)(3,8,4,7)(9,15)(10,14)(11,21)(12,18)(13,24)(16,22)(17,23)(19,20)(25,31,32,27)(26,35,29,33)(28,37,30,36)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,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, 9620533);