Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
6014 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)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,23,24,22);(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(19,24);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,22,21,23);(10,14)(13,15)(18,21)(22,23);(11,16)(13,15);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279634, #72279819, #72280725, … |
| maximal normal subgroup of | #72279634, #72279819, #72280725, … |
| maximal quotient of | #72280075, #72280147, #72280228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546801);