Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419829 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1550 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)(3,13,16,8)(4,9)(5,10,22,36)(6,24)(7,11)(12,27)(14,28,42,40)(15,29)(17,30,25,38)(18,46,45,31)(19,48,49,32)(20,39)(21,33)(23,52,51,37)(26,34)(35,44)(41,55)(43,61)(47,58)(50,59,54,62)(53,56)(57,64)(60,63);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,31)(10,35)(11,13)(12,38)(14,20)(17,27)(19,47)(21,45)(22,44)(23,53)(24,30)(26,42)(28,34)(29,36)(32,58)(33,46)(37,56)(39,40)(41,50)(43,51)(48,57)(49,64)(52,61)(54,63)(55,62)(59,60);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,32)(11,33)(12,39)(13,40)(15,41)(16,42)(17,45)(18,25)(20,27)(22,49)(23,54)(24,34)(29,55)(30,46)(31,38)(35,60)(36,48)(37,62)(43,64)(44,63)(47,53)(50,51)(52,59)(56,58)(57,61);(1,5,7,22)(2,10)(3,15,16,44)(4,19,21,49)(6,23,27,51)(8,29)(9,32)(11,36)(12,37)(13,35)(14,41,42,63)(17,43,25,53)(18,47,45,64)(20,50,26,54)(24,52)(28,55)(30,56)(31,57)(33,48)(34,59)(38,61)(39,62)(40,60)(46,58)Polycyclic presentation:
2^9|a^2=f,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286090, #72286106, #72286154, … |
| maximal normal subgroup of | #72286090, #72286106, #72286154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327026);