Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2676 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 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)(5,6)(7,10)(8,11)(9,12)(13,14,19,21,18,22,16,26)(15,23,25,17,24,20,27,28);(5,7,10,6)(8,11,9,12)(13,15,18,24)(14,20,22,23)(16,27,19,25)(17,21,28,26);(2,4)(15,24)(17,28)(20,23)(25,27);(5,8,10,9)(6,11,7,12)(13,16,18,19)(14,21,22,26)(15,25,24,27)(17,23,28,20);(5,9,10,8)(6,11,7,12)(13,17,16,23,18,28,19,20)(14,15,26,27,22,24,21,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,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, 6308058);