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) = C26×C4 |
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 | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10,11,12)(14,19)(16,17)(20,26)(21,22)(23,24)(27,28);(1,2)(3,4)(13,14,17,21,18,22,16,19)(15,20,24,27,25,28,23,26);(6,8)(13,15,18,25)(14,20,22,28)(16,23,17,24)(19,26,21,27);(2,4)(14,19)(16,17)(20,26)(21,22)(23,24)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,e^d=ei,h^d=hi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72264161, #72264214, #72264246, … |
| maximal normal subgroup of | #72264161, #72264214, #72264246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263287);