Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2799 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)(7,12)(8,10)(9,11)(13,14,19,17)(15,22,25,21)(16,18,28,20)(23,26,27,24);(2,4)(13,15)(14,21)(16,27)(17,22)(18,26)(19,25)(20,24)(23,28);(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,22)(17,18)(19,28)(21,25)(23,26)(24,27);(5,8,9,7)(6,10,11,12)(13,17,19,14)(15,23,25,27)(16,18,28,20)(21,24,22,26);(13,18,19,20)(14,16,17,28)(15,24,25,26)(21,23,22,27)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359137);