Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,9)(7,11)(10,12)(17,20)(21,22)(23,25)(24,27)(26,28);(1,2)(3,4)(5,6,8,10)(7,12,11,9)(13,14)(15,17)(16,18)(19,20);(2,4)(13,15)(14,17)(16,19)(18,20)(22,25)(23,28)(26,27)Polycyclic presentation:
2^9|b^2=g,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217697, #72217708, #72217728, … |
| maximal normal subgroup of | #72217697, #72217708, #72217728, … |
| maximal quotient of | #72215007, #72215066, #72215344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102);