Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2042 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)(5,6,8,10)(7,9,11,12)(15,21)(16,24)(18,26)(20,27)(22,28)(23,25);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,17,19)(15,22,23,18)(16,20,24,27)(21,28,25,26);(7,11)(9,12)(13,15,17,23)(14,18,19,22)(16,25,24,21)(20,28,27,26);(5,7)(6,9)(8,11)(10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,28)(24,27)(25,26);(7,11)(9,12)(14,19)(18,22)(20,27)(26,28)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6499104);