Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3511 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,17,21,18,22,23,26)(15,20,25,27,24,28,16,19);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,17)(18,24)(20,26)(21,28)(22,27)(23,25);(2,4)(6,9)(7,11)(13,16,18,25)(14,19,22,27)(15,23,24,17)(20,26,28,21);(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,23)(17,25)(18,24)(19,26)(21,27)(22,28);(6,9)(7,11)(15,24)(16,25)(19,27)(20,28)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503403);