Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148489 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11174 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,8,14,9,15,31,45)(3,16,21,48,22,44,35,37)(4,23,28,56,29,46,59,41)(5,30,6,32,7,12,34,13)(10,33,38,20,39,19,54,18)(11,25,42,26,43,27,47,60)(17,49,52,63,53,55,58,36)(24,57,51,64,50,62,61,40);(1,3)(2,10)(4,24)(5,20)(6,33)(7,18)(8,35)(9,22)(11,40)(12,44)(13,37)(14,38)(15,39)(16,30)(17,27)(19,34)(21,31)(23,55)(25,53)(26,58)(28,51)(29,50)(32,48)(36,41)(42,62)(43,64)(45,54)(46,49)(47,57)(52,60)(56,63)(59,61);(1,4,5,25,9,29,7,27)(2,11,12,41,15,43,30,56)(3,17,18,50,22,53,20,24)(6,26,8,28,34,60,31,59)(10,36,16,40,39,63,44,64)(13,46,14,47,32,23,45,42)(19,51,35,58,33,61,21,52)(37,57,54,49,48,62,38,55)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ehi,g^c=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55687);