Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,10,8,9,11,12)(14,18,19,24)(15,21)(20,27,26,25)(22,28);(13,14,15,20,16,18,22,25,17,19,21,26,23,24,28,27);(1,2,3,4)(6,9)(10,12)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,h^2=i,b^a=bh,c^a=cg,e^a=ei,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72318540, #72318553, #72318580, … |
| maximal normal subgroup of | #72318540, #72318553, #72318580, … |
| maximal quotient of | #72322457 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33730);