Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,18)(19,22)(20,21)(23,24);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(2,4)(6,8)(10,11)(14,15)(18,20)(19,23);(3,7)(6,8)(10,11)(12,16)(19,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(17,20,21,18)(19,24,23,22);(1,4,5,2)(3,8,7,6)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998953);