Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2643 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)(14,19)(20,27)(21,25)(22,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,22)(16,21)(17,19)(18,25)(20,23)(24,28)(26,27);(1,2,3,4)(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(6,10)(9,11)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22);(7,12)(9,11)(14,21)(16,18)(19,25)(20,28)(22,27)(23,26)Polycyclic presentation:
2^9|c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281624, #72281678, #72282773, … |
| maximal normal subgroup of | #72281624, #72281678, #72282773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589898);