Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139572 |
| 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 | |
491 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 491 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)(8,17)(10,15)(12,21)(16,41)(18,38,24,56)(19,51)(22,53,26,52)(23,54)(27,34,55,37)(29,49,32,47)(30,59)(31,45)(33,48,35,43)(36,44)(39,42)(40,46)(50,64,57,61)(58,62,60,63);(1,3,15,42,6,19,23,45,7,16,44,59,21,46,4,17)(2,8,20,40,12,30,36,41,13,31,54,51,28,39,10,14)(5,24,43,63,27,57,26,47,25,18,48,62,55,50,22,49)(9,32,52,64,34,60,33,38,11,29,53,61,37,58,35,56);(2,9)(3,16)(5,25)(8,29)(10,35)(11,13)(12,34)(14,38)(17,45)(19,46)(20,52)(22,26)(27,55)(28,37)(30,58)(31,32)(33,36)(39,60)(40,61)(41,56)(42,59)(43,48)(51,64)(53,54)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,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72342931, #72342991, #72342994 |
| maximal normal subgroup of | #72342931, #72342991, #72342994 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46770);