Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,22,18,21,27,23)(15,19,24,16,20,26,28,25);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,22)(17,24)(18,20)(21,26)(23,25)(27,28);(1,2,3,4)(13,16,17,26,18,25,27,19)(14,20,22,28,21,15,23,24);(7,12)(9,11)(14,21)(15,20)(22,23)(24,28);(6,10)(9,11)(15,20)(16,25)(19,26)(24,28)Polycyclic presentation:
2^9|a^2=h,c^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300548);