Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401030 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2541 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15123 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1785 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29)(16,42)(17,46)(19,34)(20,33)(22,51)(23,53)(24,38)(26,37)(28,40)(30,39)(31,45)(32,49)(35,44)(36,56)(41,47)(43,62)(48,58)(50,55)(52,57)(54,60)(59,64)(61,63);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,42)(23,52)(24,45)(26,44)(27,41)(28,48)(29,47)(30,49)(31,37)(36,59)(38,51)(39,58)(40,46)(43,55)(50,61)(53,62)(54,64)(56,63)(57,60);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,35)(17,43)(18,44)(19,22)(20,45)(27,50)(28,54)(29,55)(30,56)(32,57)(33,51)(34,42)(39,60)(40,53)(41,59)(46,61)(47,63)(48,52)(49,64)(58,62);(1,5,7,29,8,30,6,28)(2,11,13,27,14,40,12,39)(3,17,18,47,20,49,19,48)(4,23,25,55,26,56,24,54)(9,32,15,41,34,46,33,58)(10,36,21,50,38,53,37,60)(16,43,44,63,45,64,22,52)(31,57,35,59,42,61,51,62)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318785);