Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132501 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,6,12,34,46,5,27)(3,15,19,9,18,37,17,52)(4,22,24,40,59,42,61,44)(7,14,30,11,8,13,31,45)(10,33,41,26,56,29,60,25)(16,49,50,63,54,64,58,39)(20,43,32,36,21,48,28,38)(23,57,55,62,51,47,53,35);(1,3,8,21)(2,9,13,38)(4,23,25,16)(5,28,30,19)(6,32,31,17)(7,20,34,18)(10,39,22,35)(11,43,27,37)(12,15,45,48)(14,36,46,52)(24,58,33,53)(26,54,59,51)(29,55,61,50)(40,47,41,64)(42,62,56,63)(44,57,60,49);(1,4,8,25)(2,10)(3,16,21,23)(5,29,30,61)(6,33,31,24)(7,26,34,59)(9,35)(11,44)(12,41)(13,22)(14,42)(15,47)(17,53,32,58)(18,54,20,51)(19,55,28,50)(27,60)(36,63)(37,57)(38,39)(40,45)(43,49)(46,56)(48,64)(52,62)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40208);