Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,22,17,19,15,20)(16,21,25,28,24,26,23,27);(1,2,3,4)(6,9)(7,11)(14,19)(16,24)(20,22)(23,25);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,20,19,22)(16,23,24,25)(21,27,26,28);(6,9)(10,12)(14,19)(20,22)(21,26)(27,28);(13,16,18,25,17,24,15,23)(14,21,22,28,19,26,20,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,e^2=g,g^2=i,b^a=bf,d^a=df,c^b=cfi,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72311746, #72311786, #72311923, … |
| maximal normal subgroup of | #72311746, #72311786, #72311923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277704);