Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124838 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1507 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7633 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 369 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,12,22,10,7,13,23,27,5,11,21,26)(3,14,17,24,18,28,31,25,20,30,15,29,19,32,16,8);(1,3)(2,8)(4,15)(5,19)(6,18)(7,20)(9,14)(10,28)(11,29)(12,24)(13,25)(16,22)(17,23)(21,31)(26,32)(27,30);(1,4,6,22,7,23,5,21)(2,9,12,10,13,27,11,26)(3,15,18,16,20,17,19,31)(8,14,24,28,25,30,29,32)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219607, #72226635 |
| maximal normal subgroup of | #72219607, #72226635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32077);