Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132541 |
| 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 | |
527 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 527 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:
(1,2,6,12,32,44,5,26)(3,15,19,9,33,36,17,35)(4,21,24,55,59,40,58,42)(7,14,29,11,8,13,30,43)(10,22,39,25,56,28,60,23)(16,47,50,63,49,45,57,34)(18,41,31,38,20,46,27,37)(48,62,51,64,53,54,52,61);(1,3,8,20)(2,9,13,37)(4,16,23,52)(5,27,29,19)(6,31,30,17)(7,18,32,33)(10,34,21,54)(11,41,26,36)(12,15,43,46)(14,38,44,35)(22,57,24,53)(25,51,59,49)(28,50,58,48)(39,45,55,64)(40,62,56,63)(42,61,60,47);(1,4)(2,10,13,21)(3,16)(5,28)(6,22)(7,25)(8,23)(9,34,37,54)(11,42,26,60)(12,39,43,55)(14,40,44,56)(15,45,46,64)(17,53)(18,51)(19,48)(20,52)(24,30)(27,50)(29,58)(31,57)(32,59)(33,49)(35,63,38,62)(36,47,41,61)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,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, 40267);