Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499771 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4325 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)(13,14)(15,22)(16,21)(17,25)(18,20)(19,23)(24,28)(26,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(2,4)(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(14,20)(19,27)(21,25)(22,28);(6,10)(9,11)(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281290, #72281323, #72281374, … |
| maximal normal subgroup of | #72281290, #72281323, #72281374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406974);