Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,18,21)(15,22,24,27)(16,20,17,25)(19,26,28,23);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,21,28)(16,23,17,26)(20,27,25,22);(2,4)(6,9)(10,12)(13,14)(15,19)(16,25)(17,20)(18,21)(22,23)(24,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440601);