Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | |
5713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,4,8)(10,13)(14,16)(17,18)(19,21)(20,22)(23,24);(2,6)(7,8)(19,23)(21,24);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24);(1,3,5,4)(2,7,6,8)(10,13)(14,16);(1,3,5,4)(2,7,6,8)(18,22)(19,23);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72304317, #72304372 |
| maximal normal subgroup of | #72304317, #72304372 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573303);