Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3098 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)(14,19)(15,22)(16,18)(20,27)(23,28)(24,25);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,26)(17,19)(18,21)(20,25)(22,27)(24,28);(6,9)(7,11)(13,15)(14,20)(16,22)(17,25)(18,24)(19,23)(21,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,19,26)(15,24,25,22)(20,28,23,27);(6,9)(10,12)(13,16,17,18)(14,21,19,26)(15,22,25,24)(20,27,23,28)Polycyclic presentation:
2^9|a^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420180);