#9912934

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

4339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820130);