Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2112 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)(8,9)(11,12)(13,14,19,23,18,22,25,27)(15,17,21,16,24,20,28,26);(6,9)(7,11)(13,15,19,21,18,24,25,28)(14,20,23,26,22,17,27,16);(2,4)(13,16)(14,15)(17,25)(18,26)(19,20)(21,23)(22,24)(27,28);(5,7,8,11)(6,10,9,12)(13,17,18,20)(14,21,22,28)(15,23,24,27)(16,25,26,19);(14,22)(16,26)(17,20)(23,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288910);