Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×D8 | C2×C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ #1312 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
154 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,6,11,5,8)(3,12,13,16,14,7,15,9)(17,18,20,24,21,25,28,31,22,19,23,26,29,27,30,32);(2,7)(3,13)(4,5)(8,9)(10,16)(11,12)(14,15);(2,8,11,10)(3,14)(7,9,12,16)(13,15)(17,19,20,26,21,27,28,32,22,18,23,24,29,25,30,31)Polycyclic presentation:
2^9|a^2=e,c^2=ef,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286483, #72286797, #72286931, … |
| maximal normal subgroup of | #72286483, #72286797, #72286931, … |
| maximal quotient of | #72286561, #72287022, #72287200 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30157);