Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144642 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10391 |
| commutator, Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,4,8)(3,11,13,7)(5,18,17,32)(6,9,16,25)(10,26,24,20)(12,29,19,21)(14,28,31,22)(15,27,30,23);(1,3,4,13)(2,7,8,11)(5,19,17,12)(6,14,16,31)(9,22,25,28)(10,23,24,27)(15,26,30,20)(18,29,32,21);(1,4)(2,8,9,25)(3,12,14,30)(5,17)(6,16)(7,21)(10,24,18,32)(11,23)(13,19,31,15)(20,26)(22,27)(28,29)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,e^c=ei,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72235133, #72235749, C2×#144642 |
| maximal normal subgroup of | #72235133, #72235749, C2×#144642 |
| maximal quotient of | #72235171, #72235243, #72235247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51838);