Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
742 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)(5,6,10,11,9,7,8,12)(14,18,20,25)(19,21,26,27);(5,7,10,12,9,6,8,11)(13,14,16,25,17,20,22,18)(15,21,23,19,24,27,28,26);(1,2,3,4)(5,8,9,10)(6,11,7,12);(13,15,22,28,17,24,16,23)(14,19,25,27,20,26,18,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,e^b=eh,g^b=gh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138618);