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 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,21)(17,20)(18,22)(19,24)(25,28)(26,27);(5,7)(6,9)(8,12)(10,11)(13,15,17,24)(14,19,20,23)(16,18,26,25)(21,28,27,22);(1,2,3,4)(13,15,18,16,17,24,25,26)(14,19,22,27,20,23,28,21);(6,10)(9,11)(14,20)(19,23)(21,27)(22,28);(7,12)(9,11)(13,16,25,15,17,26,18,24)(14,21,28,23,20,27,22,19)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,e^2=hi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721577);