Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#2041 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1169 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)(6,31)(7,14)(8,13)(9,16)(10,20)(11,27)(15,33)(18,50)(19,40)(21,37)(22,38)(23,39)(25,36)(26,44)(28,42)(29,61)(30,46)(32,43)(34,45)(35,60)(41,54)(47,56)(48,55)(49,53)(51,64)(52,62)(57,63)(58,59);(1,3)(2,10)(4,25)(5,19)(6,22)(7,21)(8,20)(9,23)(11,41)(12,17)(13,39)(14,47)(15,48)(16,40)(18,28)(24,59)(26,53)(27,52)(29,57)(30,55)(31,37)(32,54)(33,56)(34,51)(35,58)(36,44)(38,46)(42,62)(43,64)(45,50)(49,61)(60,63);(1,4,5,26,9,29,8,35)(2,11,13,43,16,45,12,42)(3,18,20,52,23,54,19,51)(6,32,30,27,33,28,7,34)(10,36,17,49,40,63,39,59)(14,44,15,24,46,60,31,61)(21,57,56,53,55,25,22,58)(37,41,38,62,48,50,47,64);(1,5,9,8)(2,12,16,13)(3,19,23,20)(4,26,29,35)(6,30,33,7)(10,17,40,39)(11,42,45,43)(14,31,46,15)(18,51,54,52)(21,22,55,56)(24,44,61,60)(25,53,57,58)(27,28,34,32)(36,49,63,59)(37,38,48,47)(41,62,50,64)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=ghi,h^b=hi,f^c=fhi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249769, #72249772, #72249831, … |
| maximal normal subgroup of | #72249769, #72249772, #72249831, … |
| maximal quotient of | #72250034, C2×#499091, #72323030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406287);