Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35256 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(14,19)(15,17)(20,22)(21,26)(23,25)(27,28);(1,2)(3,4)(6,9)(10,11)(13,14,15,20,18,22,17,19)(16,21,23,27,24,28,25,26);(2,4)(5,7,8,12)(6,10,9,11)(13,15,18,17)(14,20,22,19)(16,23,24,25)(21,27,28,26);(2,4)(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,27)(22,28);(16,24)(21,28)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354843);