Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139571 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1910 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #9469 |
| prime radical | #532 | D4 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 311 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:
(1,2,6,12,7,14,22,13)(3,15,19,44,20,46,49,45)(4,21,16,10,24,39,48,40)(5,11,28,41,26,9,29,37)(8,30,33,51,35,59,34,17)(18,43,54,64,25,42,55,63)(23,57,52,38,27,56,47,36)(31,58,60,50,32,61,62,53);(1,3,16,30,6,19,24,51,7,20,48,59,22,49,4,17)(2,8,21,45,13,34,40,46,14,35,39,44,12,33,10,15)(5,25,47,61,29,54,27,50,26,18,52,58,28,55,23,53)(9,31,56,63,37,60,36,43,11,32,57,64,41,62,38,42);(2,9)(5,26)(8,31)(10,38)(11,14)(12,41)(13,37)(15,42)(18,25)(21,56)(23,27)(28,29)(32,35)(33,62)(34,60)(36,40)(39,57)(43,46)(44,64)(45,63)(47,52)(50,53)(54,55)(58,61)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=gi,e^2=i,f^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 | #72320761, #72342994, #72343007 |
| maximal normal subgroup of | #72320761, #72342994, #72343007 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46777);