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 | #33556 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,18,39)(4,19,22,49)(5,23,15,10)(6,11,26,36)(8,27,30,16)(9,31,33,21)(14,40,44,63)(17,38,25,55)(20,50,41,48)(24,53,45,34)(28,58,59,64)(29,54,35,46)(32,51,52,60)(37,56,62,43)(42,47,57,61);(1,3,5,16,7,18,15,27)(2,8,10,13,12,30,23,39)(4,14,20,42,22,44,41,57)(6,25,24,54,26,17,45,46)(9,28,32,37,33,59,52,62)(11,35,34,55,36,29,53,38)(19,47,48,40,49,61,50,63)(21,43,51,64,31,56,60,58);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,28)(10,32)(11,19)(12,33)(13,37)(15,41)(16,42)(17,43)(18,44)(23,52)(24,51)(25,56)(26,31)(27,57)(29,47)(30,59)(34,48)(35,61)(36,49)(38,40)(39,62)(45,60)(46,64)(50,53)(54,58)(55,63);(1,3,15,27,7,18,5,16)(2,8,23,39,12,30,10,13)(4,14,41,57,22,44,20,42)(6,17,45,54,26,25,24,46)(9,28,52,62,33,59,32,37)(11,29,53,55,36,35,34,38)(19,47,50,63,49,61,48,40)(21,43,60,58,31,56,51,64)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=ei,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 | #72312282, #72312376, #72312405, … |
| maximal normal subgroup of | #72312282, #72312376, #72312405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408002);