Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1325 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1325 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,7,10,8,11,12,9)(14,17,18,20)(21,22)(23,27,28,25)(24,26);(1,2,3,4)(6,9)(7,12)(10,11)(15,19)(17,20)(21,22)(23,25)(24,26)(27,28);(13,14)(15,20)(16,18)(17,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=hi,f^2=g,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72217679, #72217683, #72279350, … |
| maximal normal subgroup of | #72217679, #72217683, #72279350, … |
| maximal quotient of | #72215007, #72215020, #72215055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6835);