#9863312

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

4503 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)(5,6)(7,8)(9,10,14,16)(11,18,21,15)(12,13,22,17)(19,23,24,20)(25,26)(27,33)(28,29)(30,34)(31,40)(32,36)(35,37)(38,39);(2,6)(3,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,27)(26,33)(28,35)(29,37)(30,40)(31,34)(32,38)(36,39);(1,3,5,7)(2,4,6,8)(10,16)(12,22)(15,18)(19,24)(25,28,32,30)(26,34,36,29)(27,35,38,40)(31,39,37,33);(1,4)(2,7)(3,6)(5,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,23)(18,20)(21,24)(25,29,32,34)(26,28,36,30)(27,37,38,31)(33,35,39,40);(9,11)(10,18)(12,19)(13,23)(14,21)(15,16)(17,20)(22,24)(25,30,32,28)(26,34,36,29)(27,35,38,40)(31,33,37,39);(9,13,14,17)(10,12,16,22)(11,20,21,23)(15,19,18,24)(25,31,32,37)(26,35,36,40)(27,34,38,29)(28,39,30,33)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=ghi,e^2=ghi,f^2=h,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, 9770508);