Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1698 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)(5,6,7,8)(13,14,18,21,19,22,16,17)(15,23,24,20,27,28,26,25);(9,10,11,12)(13,15)(14,20)(16,24)(17,28)(18,26)(19,27)(21,23)(22,25);(2,4)(13,16,19,18)(14,17,22,21)(15,24,27,26)(20,28,25,23);(13,17,16,22,19,21,18,14)(15,25,26,28,27,20,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72265203, #72265213, #72265262, … |
| maximal normal subgroup of | #72265203, #72265213, #72265262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263415);