Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35831 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2043 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)(13,14,19,20)(15,23,27,21)(16,18,22,17)(24,26,25,28);(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,21,20,23)(16,26,22,28)(17,25,18,24);(1,2,3,4)(13,16)(14,18)(15,24)(17,20)(19,22)(21,28)(23,26)(25,27);(6,10)(9,11)(13,17)(14,22)(15,25)(16,20)(18,19)(21,28)(23,26)(24,27);(7,12)(9,11)(13,18,19,17)(14,22,20,16)(15,26,27,28)(21,24,23,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590023);