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 | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,16,18,22,17,20)(15,23,27,28,26,21,25,24);(1,2)(3,4)(14,20,22,16)(21,28,23,24);(6,9)(10,12)(13,15,19,27,18,26,17,25)(14,21,16,24,22,23,20,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,14,18,20,19,22)(15,24,25,21,26,28,27,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=fg,e^2=i,g^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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264065);