#9947522

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

4587 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)(3,6)(4,5)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,40,36,39);(2,5)(6,7)(12,19)(15,17)(20,24)(22,23)(25,26,29,30)(27,31,34,38)(28,32,37,35)(33,39,36,40);(10,11)(15,17)(16,18)(22,23)(26,30)(31,38)(32,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,2)(3,7)(4,5)(6,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,3,4,8)(2,6,5,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,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, 9854718);