#9946993

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

4507 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)(3,8)(4,6)(5,7)(9,10,15,16)(11,14,12,19)(13,17,22,24)(18,23,20,21)(25,26,31,32)(27,34,29,28)(30,38,39,33)(35,36,37,40);(1,3)(2,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,18)(16,19)(17,23)(20,22)(21,24)(25,27,31,29)(26,28,32,34)(30,35,39,37)(33,36,38,40);(10,16)(11,12)(17,24)(18,20)(26,32)(28,34)(33,38)(36,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,23,24,21)(25,28)(26,27)(29,32)(30,36)(31,34)(33,35)(37,38)(39,40);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,18)(12,20)(14,23)(15,22)(16,24)(19,21)(25,29,31,27)(26,28,32,34)(30,35,39,37)(33,40,38,36);(1,2)(3,6)(4,8)(5,7)(9,14)(10,11)(12,16)(13,21)(15,19)(17,20)(18,24)(22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,38)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^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, 9854189);