Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,7,10,8,11,12,9)(13,14,17,21,18,22,26,28,16,19,23,27,25,15,20,24);(1,2,3,4)(6,9)(7,12)(10,11)(14,19)(15,22)(21,27)(24,28);(13,15,23,28,18,14,20,27,16,22,17,24,25,19,26,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=h,h^2=i,b^a=bd,c^b=ci,d^b=dg,e^b=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225365, #72225377, D8×#346, … |
| maximal normal subgroup of | #72225365, #72225377, D8×#346, … |
| maximal quotient of | #72225363, #72225365, #72225377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36336);