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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(10,14,17,16)(15,21,23,22);(5,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24);(2,4)(9,11,12,18,13,19,20,24)(10,15,16,22,17,23,14,21);(10,16,17,14)(15,22,23,21)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,d^2=g,e^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 | #72216212 |
| maximal normal subgroup of | #72216212 |
| maximal quotient of | #72216085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263944);