Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,20,16)(17,21,25,28)(23,26,27,24);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,22)(17,24)(18,20)(21,23)(25,26)(27,28);(2,4)(7,12)(9,11)(13,16,22,15,18,19,14,20)(17,24,28,23,25,26,21,27);(6,10)(9,11)(13,14,18,22)(15,19,20,16)(17,21,25,28)(23,26,27,24);(13,17,14,21,18,25,22,28)(15,23,16,24,20,27,19,26)Polycyclic presentation:
2^9|a^2=i,c^2=fhi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308331 |
| maximal normal subgroup of | #72308331 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717080);