Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9468 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,14)(4,20)(5,11,25,9)(6,28)(7,13)(8,16)(10,15)(12,21)(17,40,24,38)(18,52)(19,42)(22,54,26,53)(23,55)(27,34,56,37)(29,47,30,50)(31,60)(32,48)(33,49,35,44)(36,45)(39,43)(41,46)(51,61,57,62)(58,64,59,63);(1,3,15,43,6,18,23,48,7,19,45,60,21,46,4,16)(2,8,20,41,12,31,36,42,13,32,55,52,28,39,10,14)(5,24,44,64,27,57,26,47,25,17,49,63,56,51,22,50)(9,29,53,61,34,58,33,40,11,30,54,62,37,59,35,38);(2,9)(5,25)(8,29)(10,35)(11,13)(12,34)(14,38)(17,24)(20,53)(22,26)(27,56)(28,37)(30,32)(31,58)(33,36)(39,59)(40,42)(41,61)(44,49)(47,50)(51,57)(52,62)(54,55)(63,64)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72342902, #72342994, #72343007 |
| maximal normal subgroup of | #72342902, #72342994, #72343007 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46769);