Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,23,18,20,16,22)(15,21,25,27,24,28,26,19);(2,3)(13,15,18,24)(14,19,20,27)(16,25,17,26)(21,23,28,22);(1,2,4,3)(6,9)(10,12)(14,20)(15,24)(22,23)(25,26);(1,3,4,2)(13,15)(14,21)(16,26)(17,25)(18,24)(19,22)(20,28)(23,27);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,22,20,23)(15,25,24,26)(19,21,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374623);