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 | |
3181 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)(13,14)(15,21)(16,20)(17,22)(18,19)(23,28)(24,26)(25,27);(1,2)(3,4)(6,9)(10,11)(13,15,21,14,18,22,17,19)(16,24,28,27,25,23,26,20);(2,4)(5,7,8,12)(6,10,9,11)(14,19)(15,22)(20,27)(23,24);(2,4)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,28)(22,24);(16,25)(20,27)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,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, 6354845);