Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #11384 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13,16,42)(4,18)(5,22)(6,27)(7,11)(8,15,31,48)(9,20)(10,25)(12,29)(14,43,45,41)(17,49,28,36)(19,51)(21,35)(23,32,57,39)(24,58)(26,40)(30,44,33,47)(34,59)(37,53)(38,60)(46,63,55,64)(50,54)(52,61,56,62);(1,3)(2,8,11,31)(4,14)(5,23)(6,28)(7,16)(9,33,18,30)(10,36,38,49)(12,39,27,32)(13,40,42,22)(15,25)(17,29)(19,52)(20,45)(21,55)(24,56)(26,57)(34,63,37,64)(35,61,50,62)(41,51,43,58)(44,53)(46,54)(47,59)(48,60);(1,4)(2,9,11,18)(3,14)(5,24)(6,21)(7,20)(8,30,31,33)(10,37,38,34)(12,35,27,50)(13,41,42,43)(15,47)(16,45)(17,46)(19,26)(22,51,40,58)(23,56)(25,59)(28,55)(29,54)(32,61,39,62)(36,63,49,64)(44,48)(52,57)(53,60)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279962, #72280652, #72307204, … |
| maximal normal subgroup of | #72279962, #72280652, #72307204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56666);