Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #33555 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1009 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)(3,15)(4,19)(5,9)(6,26)(7,13)(8,14)(10,22)(11,18)(12,28)(16,43)(17,42)(20,47)(21,53)(23,50)(24,32)(25,33)(27,59)(29,37)(30,40)(31,51)(34,60)(35,62)(36,56)(38,46)(39,57)(41,44)(45,48)(49,55)(52,63)(54,58)(61,64);(1,3,6,5,8,18,28,25)(2,9,12,15,14,33,26,11)(4,16,21,20,23,45,55,52)(7,29,27,57,30,17,46,24)(10,31,35,41,36,61,58,34)(13,39,38,37,40,32,59,42)(19,47,49,43,50,63,53,48)(22,44,54,51,56,60,62,64);(1,4)(2,10)(3,16)(5,20)(6,21)(7,22)(8,23)(9,31)(11,34)(12,35)(13,19)(14,36)(15,41)(17,44)(18,45)(24,51)(25,52)(26,58)(27,54)(28,55)(29,60)(30,56)(32,47)(33,61)(37,48)(38,49)(39,63)(40,50)(42,43)(46,62)(53,59)(57,64);(1,5,6,18,8,25,28,3)(2,11,12,9,14,15,26,33)(4,20,21,45,23,52,55,16)(7,24,27,29,30,57,46,17)(10,34,35,31,36,41,58,61)(13,37,38,32,40,42,59,39)(19,48,49,47,50,43,53,63)(22,51,54,60,56,64,62,44)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312286, #72312372, #72312405, … |
| maximal normal subgroup of | #72312286, #72312372, #72312405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407999);