Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#427 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #20818 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #405 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9,17,33)(4,20,22,35)(5,11)(6,14)(7,29)(8,30)(10,23,36,54)(12,27)(13,26)(15,40)(16,32,25,38)(18,47,46,60)(19,48,45,34)(21,49,53,37)(24,55,52,50)(28,39)(31,56)(41,57)(42,59)(43,58)(44,63)(51,64)(61,62);(1,3)(2,9)(4,15)(5,25)(6,17)(7,18)(8,19)(10,31)(11,38)(12,32)(13,34)(14,33)(16,27)(20,40)(21,51)(22,42)(23,43)(24,44)(26,45)(28,46)(29,47)(30,48)(35,59)(36,58)(37,57)(39,60)(41,53)(49,64)(50,61)(52,62)(54,56)(55,63);(1,4,7,23,8,24,27,53,6,22,28,54,26,52,5,21)(2,10,12,20,13,37,29,50,14,36,11,35,30,49,39,55)(3,15,18,43,19,44,16,41,17,42,46,56,45,62,25,51)(9,31,32,40,34,57,47,61,33,58,38,59,48,64,60,63);(1,5,26,28,6,27,8,7)(2,11,13,39,14,12,30,29)(3,16,45,18,17,25,19,46)(4,21,52,54,22,53,24,23)(9,32,34,47,33,38,48,60)(10,35,37,55,36,20,49,50)(15,41,62,43,42,51,44,56)(31,40,57,61,58,59,64,63)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=eh,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217287, #72217290, #72224951, … |
| maximal normal subgroup of | #72217287, #72217290, #72224951, … |
| maximal quotient of | C2×#494212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401408);