#9847861

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

4473 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9755057);