Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1502 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7621 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,6,12,22,30,8,13,21,10,7,24,4,20)(3,14,16,9,18,26,15,28,19,31,23,29,17,25,32,27);(1,3)(2,9)(4,15)(5,23)(6,18)(7,17)(8,19)(10,25)(11,26)(12,28)(13,29)(14,20)(16,21)(22,32)(24,27)(30,31);(1,4,6,5,8,22,7,21)(2,10,24,30,13,11,12,20)(3,15,18,23,19,32,17,16)(9,25,27,31,29,26,28,14)Polycyclic presentation:
2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31986);