Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2996 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 2996 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:
(1,2)(3,4)(5,6,10,14)(7,11,17,24)(8,12,20,27)(9,13,21,28)(15,25,18,22)(16,23,30,34)(19,26,32,36)(29,35,31,33);(5,7)(6,11)(8,18,20,15)(9,16)(10,17)(12,25,27,22)(13,23)(14,24)(19,29,32,31)(21,30)(26,33,36,35)(28,34);(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,26)(14,27)(16,29)(17,18)(21,32)(23,33)(24,25)(28,36)(30,31)(34,35);(2,4)(5,9)(6,13)(7,16)(8,19)(10,21)(11,23)(12,26)(14,28)(15,29)(17,30)(18,31)(20,32)(22,33)(24,34)(25,35)(27,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252227, #72261220, #72283486, … |
| maximal normal subgroup of | #72252227, #72261220, #72283486, … |
| maximal quotient of | #72252216, #72261273, #72261459 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325021);