Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2470 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)(13,14,18,15,17,20,26,22)(16,19,25,21,24,27,28,23);(5,7)(6,9)(8,12)(10,11)(13,15,18,20,17,22,26,14)(16,23,25,19,24,21,28,27);(1,2,3,4)(7,12)(9,11)(13,16)(14,19)(15,21)(17,24)(18,25)(20,27)(22,23)(26,28);(6,10)(9,11);(16,24)(19,27)(21,23)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72312809, #72312818, #72312933 |
| maximal normal subgroup of | #72312809, #72312818, #72312933 |
| maximal quotient of | #72315823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283331);