Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,20,23,28)(16,21,17,19)(24,26,25,27);(5,7)(6,9)(8,11)(10,12)(14,19)(15,23)(16,17)(20,27)(21,22)(26,28);(1,2,3,4)(7,11)(9,12)(13,15)(14,20)(16,25)(17,24)(18,23)(19,26)(21,27)(22,28);(7,11)(9,12)(13,16,18,17)(14,21,22,19)(15,24,23,25)(20,26,28,27);(6,10)(9,12)(14,22)(19,21)(20,28)(26,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380264);