Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
5352 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)(9,10,13,16)(11,15,12,14);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(18,21)(19,23);(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(17,19,20,23)(18,22,21,24);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(18,21)(22,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304029 |
| maximal normal subgroup of | #72304029 |
| maximal quotient of | #72304112 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7608935);