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 | #39342 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,24,20,22);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,22,23,24);(10,13)(14,15)(19,20)(22,24);(9,11,12,16)(10,14,13,15);(1,3,5,4)(2,7,8,6)(17,20,21,19)(18,22,23,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308682, #72308792 |
| maximal normal subgroup of | #72308682, #72308792 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7545692);