Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3549 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3616 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)(13,14,20,23,19,16,18,24)(15,22,25,27,17,21,26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,22)(17,19)(18,26)(20,25)(23,28)(24,27);(1,2,3,4)(13,16,19,14)(15,22,17,21)(18,24,20,23)(25,28,26,27);(7,12)(9,11)(13,17,19,15)(14,22,16,21)(18,25,20,26)(23,27,24,28);(6,10)(9,11)(13,18,19,20)(14,23,16,24)(15,25,17,26)(21,27,22,28)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300636);