Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(14,19,21,22)(15,24)(16,26)(17,25)(20,27,23,28);(2,4)(6,7)(11,12)(13,14,16,22,17,23,26,28,18,21,15,19,25,20,24,27);(5,7,9,12,8,6,10,11)(14,20,21,23)(15,16)(19,27,22,28)(24,26)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,c^b=cgh,e^b=ei,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225063, #72225064, #72225106, … |
| maximal normal subgroup of | #72225063, #72225064, #72225106, … |
| maximal quotient of | #72225060, #72225082, #72286508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35915);