Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,2,3,4)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(6,10)(9,11)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28);(7,12)(9,11)(13,15,16,23,18,25,17,24)(14,19,26,28,21,27,20,22)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722038);