Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2265 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)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14)(15,24)(16,21)(17,27)(18,28)(19,23)(20,26)(22,25);(2,4)(7,12)(9,11)(13,15,18,26,19,17,16,25)(14,20,21,24,23,22,28,27);(6,10)(9,11)(13,16,19,18)(14,21,23,28)(15,25,17,26)(20,24,22,27);(13,17,18,25,19,15,16,26)(14,22,21,27,23,20,28,24)Polycyclic presentation:
2^9|c^2=fh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257193, #72257196, #72257204, … |
| maximal normal subgroup of | #72257193, #72257196, #72257204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717053);