Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1667 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1667 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
503 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)(3,17)(4,24)(5,12)(6,14)(7,13)(8,15)(9,16)(10,20)(11,26)(18,46)(19,44)(21,33)(22,34)(23,45)(25,54)(27,55)(28,40)(29,41)(30,37)(31,42)(32,53)(35,51)(36,50)(38,57)(39,56)(43,48)(47,59)(49,58)(52,61)(60,64)(62,63);(1,3,7,21,5,19,28,50,9,23,29,51,8,22,6,20)(2,10,14,34,15,35,41,45,16,36,40,44,12,33,13,17)(4,18,30,52,25,47,56,63,27,49,57,64,31,53,26,48)(11,32,42,60,38,58,55,62,39,59,54,61,37,46,24,43);(1,4)(2,11)(3,18)(5,25)(6,26)(7,30)(8,31)(9,27)(10,32)(12,37)(13,24)(14,42)(15,38)(16,39)(17,43)(19,47)(20,48)(21,52)(22,53)(23,49)(28,56)(29,57)(33,46)(34,60)(35,58)(36,59)(40,54)(41,55)(44,61)(45,62)(50,63)(51,64);(1,5,9,8)(2,12,16,15)(3,19,23,22)(4,25,27,31)(6,7,28,29)(10,33,36,35)(11,37,39,38)(13,40,41,14)(17,44,45,34)(18,47,49,53)(20,21,50,51)(24,54,55,42)(26,30,56,57)(32,46,59,58)(43,61,62,60)(48,52,63,64)Polycyclic presentation:
2^9|b^2=eg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225389, #72225394, #72225403, … |
| maximal normal subgroup of | #72225389, #72225394, #72225403, … |
| maximal quotient of | #72225448 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397680);