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 | |
3358 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 3358 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,12,23,28)(7,15,17,30,18,11,22,26)(8,19,21,27,20,31,24,13)(16,32,34,25,33,36,35,29);(1,2)(3,4)(5,7,10,18)(6,11,12,15)(8,16,20,33)(9,22,23,17)(13,29,27,25)(14,30,28,26)(19,36,31,32)(21,35,24,34);(2,4)(6,12)(9,23)(11,15)(13,27)(17,22)(21,24)(25,29)(34,35);(5,8)(6,13)(7,16)(9,24)(10,20)(11,25)(12,27)(14,31)(15,29)(17,35)(18,33)(19,28)(21,23)(22,34)(26,32)(30,36)Polycyclic presentation:
2^9|a^2=h,b^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, #72263056, #72263057, … |
| maximal normal subgroup of | #72262999, #72263056, #72263057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356689);