Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500596 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(6,13)(7,19)(8,25)(9,11)(10,26)(14,31)(15,30)(17,18)(20,35)(21,24)(22,34)(23,28)(29,32)(33,36);(1,2,3,4)(5,6,9,13,12,18,11,17)(7,20,23,33,24,29,22,14)(8,15,25,16,26,30,10,27)(19,31,34,32,21,36,28,35);(2,4)(5,7)(6,14)(8,21)(9,22)(10,28)(11,23)(12,24)(13,29)(15,32)(16,31)(17,20)(18,33)(19,26)(25,34)(27,36)(30,35);(5,8,11,10,12,26,9,25)(6,15,17,27,18,30,13,16)(7,21,23,28,24,19,22,34)(14,32,20,36,33,35,29,31)Polycyclic presentation:
2^9|b^2=g,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239803, #72239816, #72239821, … |
| maximal normal subgroup of | #72239803, #72239816, #72239821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413495);