#9794253

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=h,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, 9701449);