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 | #52165 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2137 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)(13,14,19,16,18,21,22,15)(17,25,27,23,26,20,28,24);(5,7)(6,9)(8,12)(10,11)(13,15,19,14,18,16,22,21)(17,24,27,25,26,23,28,20);(2,4)(13,16,19,21,18,15,22,14)(17,24,27,25,26,23,28,20);(7,12)(9,11)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28);(6,10)(9,11)(14,21)(15,16)(17,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301953);