Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3232 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3232 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,15,25,31)(7,13,18,29,8,12,20,27)(9,21,23,30,22,32,24,14)(17,33,34,26,19,36,35,28);(1,2)(3,4)(5,7,11,8)(6,12,15,13)(9,17,22,19)(10,20,25,18)(14,28,30,26)(16,29,31,27)(21,36,32,33)(23,35,24,34);(2,4)(5,8,11,7)(6,13,15,12)(9,19,22,17)(10,18,25,20)(14,28,30,26)(16,27,31,29)(21,36,32,33)(23,34,24,35);(5,9)(6,14)(7,17)(8,19)(10,24)(11,22)(12,26)(13,28)(15,30)(16,32)(18,35)(20,34)(21,31)(23,25)(27,33)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262999, #72263055, #72267103 |
| maximal normal subgroup of | #72262999, #72263055, #72267103 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356690);