Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| 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 |
| prime radical | #24210 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,26)(18,20)(19,25)(23,28)(24,27);(2,3)(13,15)(14,19)(16,25)(17,24)(18,23)(20,27)(21,28)(22,26);(1,2,4,3)(6,9)(10,12)(14,20)(15,23)(21,26)(24,25);(1,3,4,2)(13,16,18,17)(14,21,20,26)(15,24,23,25)(19,22,27,28);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,26)(22,28)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257067, #72257113, #72257176, … |
| maximal normal subgroup of | #72257067, #72257113, #72257176, … |
| maximal quotient of | #72265568 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374344);