Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2610 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)(7,10)(8,9)(11,12)(13,14,18,21,19,22,17,16)(15,23,25,28,26,24,27,20);(2,4)(13,15)(14,20)(16,23)(17,25)(18,27)(19,26)(21,24)(22,28);(6,9)(10,12)(13,14,18,21,19,22,17,16)(15,24,25,20,26,23,27,28);(13,16,17,22,19,21,18,14)(15,20,27,24,26,28,25,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,16,22,21)(15,25,26,27)(20,23,28,24)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374646);