#9947281

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

4455 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,e^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, 9854477);