Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52163 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,15,17,20,26,22)(16,19,25,21,24,23,27,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,15,17,20,26,22)(16,23,25,28,24,19,27,21);(2,4)(13,15,18,20,17,22,26,14)(16,21,25,23,24,28,27,19);(7,12)(9,11)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(6,10)(9,11)(16,24)(19,23)(21,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301937);