Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5372 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)(3,7)(4,6)(5,8)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,6,8,7)(9,10,12,14)(11,15,16,13)(18,19)(20,24);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22);(1,3,5,4)(2,7,8,6)(10,14)(11,16)(17,20,21,24)(18,22,19,23);(11,16)(13,15)(18,19)(22,23);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7612895);