Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2615 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,15,18,22,17,21)(16,20,24,23,25,27,26,28);(6,9)(7,11)(13,15,19,22,18,21,17,14)(16,23,24,27,25,28,26,20);(1,2,3,4)(13,16,19,24,18,25,17,26)(14,20,15,23,22,27,21,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,15)(16,24,25,26)(20,23,27,28);(6,9)(10,12)(14,22)(15,21)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257335, #72283105, #72283123, … |
| maximal normal subgroup of | #72257335, #72283105, #72283123, … |
| maximal quotient of | #72257830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303432);