Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42928 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1800 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)(15,24)(16,20)(17,22)(18,21)(19,23)(25,28)(26,27);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,19)(18,22)(20,27)(21,24)(23,28)(25,26);(1,2,3,4)(6,10)(7,12)(13,16,26,22,18,25,19,15)(14,20,27,17,21,28,23,24);(6,10)(9,11)(14,21)(17,24)(20,28)(23,27);(13,17,26,28,18,24,19,20)(14,22,27,25,21,15,23,16)Polycyclic presentation:
2^9|b^2=fi,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723431);