Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418858 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1883 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)(4,18,21,9)(5,10)(6,12)(7,11)(14,42)(15,30,45,35)(17,39,25,31)(19,49)(20,50,53,33)(22,40,26,34)(23,37)(24,36)(27,41)(28,38)(29,44)(32,47)(43,56)(46,57,48,58)(51,59)(52,64)(54,61,55,60)(62,63);(1,3)(2,8)(4,19)(5,15)(6,25)(7,16)(9,32)(10,35)(11,13)(12,39)(14,22)(17,28)(18,49)(20,52)(21,47)(23,48)(24,45)(26,44)(27,46)(29,34)(30,36)(31,38)(33,59)(37,57)(40,42)(41,58)(43,54)(50,64)(51,53)(55,63)(56,61)(60,62);(1,4)(2,9)(3,14)(5,20)(6,26)(7,21)(8,29)(10,33)(11,18)(12,40)(13,42)(15,43)(16,44)(17,47)(19,25)(22,28)(23,55)(24,53)(27,54)(30,56)(31,49)(32,39)(34,38)(35,62)(36,50)(37,61)(41,60)(45,63)(46,51)(48,52)(57,59)(58,64);(1,5,7,24)(2,10,11,36)(3,15,16,45)(4,20,21,53)(6,27,28,23)(8,30,13,35)(9,33,18,50)(12,41,38,37)(14,43,44,63)(17,48,25,46)(19,51,47,52)(22,55,26,54)(29,56,42,62)(31,58,39,57)(32,59,49,64)(34,61,40,60)Polycyclic presentation:
2^9|a^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326055);