Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 |
| prime radical | #39958 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,12,16)(18,22)(23,24);(1,3,5,4)(2,7,8,6)(9,12,14,10)(11,16,13,15)(17,18)(19,23)(20,24)(21,22);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,15,12,16)(17,19,21,20)(18,23,22,24);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,23,22,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7545359);