Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10)(11,12)(13,14,18,23)(15,24,27,28)(16,22,17,21)(19,26,20,25);(5,6)(7,8)(13,15,17,26,18,27,16,25)(14,19,22,28,23,20,21,24);(2,4)(6,8);(10,12)(13,15,16,25,18,27,17,26)(14,20,21,28,23,19,22,24)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264150, #72264152, #72264257, … |
| maximal normal subgroup of | #72264150, #72264152, #72264257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302355);