#9947474

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854670);