#9812912

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

4361 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=g,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, 9720108);