Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458125 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,18)(8,22)(9,27)(10,15)(11,17)(12,16)(13,20)(14,26)(19,31)(21,30)(23,32)(24,29)(25,34)(28,33)(35,36);(1,2,4,3)(5,7,11,21,10,20,12,9)(6,13,17,27,15,18,16,30)(8,23,26,35,25,24,28,19)(14,31,34,32,33,36,22,29);(2,3)(5,8)(6,14)(7,19)(9,23)(10,25)(11,28)(12,26)(13,29)(15,33)(16,34)(17,22)(18,32)(20,35)(21,24)(27,36)(30,31);(1,3,4,2)(5,9,12,20,10,21,11,7)(6,13,17,27,15,18,16,30)(8,24,26,19,25,23,28,35)(14,32,22,31,33,29,34,36)Polycyclic presentation:
2^9|b^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382956);