Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,7)(13,14,16,21,17,19,24,18)(15,20,22,27,23,26,28,25);(9,10,11,12)(14,18)(15,22)(16,24)(19,21)(23,28)(25,27);(2,4)(5,6,8,7)(14,19)(15,23)(18,21)(22,28);(5,7,8,6)(13,15,16,22,17,23,24,28)(14,20,21,27,19,26,18,25)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=efg,d^2=fgi,e^2=i,g^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264234);