#9947327

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

4333 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)(3,4)(5,8)(6,7)(9,10,15,17)(11,20,14,16)(12,18,22,24)(13,19,23,21)(27,33)(30,34)(35,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,18,23,24)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(1,2,5,8)(3,7,6,4)(10,17)(12,22)(16,20)(19,21)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4)(2,6)(3,8)(5,7)(9,12)(10,18)(11,19)(13,16)(14,21)(15,22)(17,24)(20,23)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,19)(11,18)(12,16)(14,24)(15,23)(17,21)(20,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3)(2,7)(4,8)(5,6)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,18,23,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=ghi,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, 9854523);