Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,7,8)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,22)(21,23);(2,3)(9,11,13,12)(10,14,16,15)(17,19)(18,22)(20,21)(23,24);(5,6)(7,8)(17,18)(19,24)(20,22)(21,23);(2,3)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,24)(21,23);(1,3,4,2)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72329311, #72329330, #72329393, … |
| maximal normal subgroup of | #72329311, #72329330, #72329393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 750743);