Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| 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 | #59680 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,27)(22,24)(23,25)(26,28);(2,3)(6,9)(10,11)(14,18)(15,21)(17,23)(19,26)(22,28)(24,27);(1,2,4,3);(1,3,4,2)(13,15,17,21,16,22,23,28)(14,19,20,27,18,24,25,26);(5,7,8,12)(6,10,9,11)(14,18)(19,24)(20,25)(26,27)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308452, #72308472 |
| maximal normal subgroup of | #72308452, #72308472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520970);