#9794263

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

4329 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)(3,4)(5,6)(7,8)(9,10,16,13)(11,14,17,12)(15,21,23,18)(19,20,24,22)(26,32)(27,30)(28,34)(37,40);(2,6)(4,7)(9,11)(10,14)(12,13)(15,19)(16,17)(18,22)(20,21)(23,24)(25,26,31,32)(27,33,30,35)(28,29,34,39)(36,40,38,37);(9,12,16,14)(10,17,13,11)(15,20,23,22)(18,24,21,19)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,39,40);(1,3)(2,7)(4,6)(5,8)(9,13,16,10)(11,12,17,14)(15,21,23,18)(19,20,24,22)(25,28,31,34)(26,29,32,39)(27,36,30,38)(33,40,35,37);(1,4,5,7)(2,3,6,8)(9,14,16,12)(10,17,13,11)(15,20,23,22)(18,19,21,24)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,36)(35,38);(1,3)(2,4)(5,8)(6,7)(9,15,16,23)(10,18,13,21)(11,19,17,24)(12,20,14,22)(25,30,31,27)(26,35,32,33)(28,38,34,36)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=gh,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, 9701459);