Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#546 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,7,8)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(2,4)(9,10)(11,12)(13,15,18,26)(14,19,22,28)(16,25,17,24)(20,27,21,23);(6,8)(10,12)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(14,21)(16,17)(19,23)(20,22)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=efi,b^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264472, #72264480, #72264497, … |
| maximal normal subgroup of | #72264472, #72264480, #72264497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302657);