Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401030 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2541 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15124 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1999 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)(4,10)(5,25)(6,13)(7,12)(8,14)(9,18)(11,27)(16,42)(17,31)(19,33)(20,32)(21,30)(22,50)(23,37)(24,38)(26,40)(28,39)(29,36)(34,44)(35,51)(41,47)(43,62)(45,53)(46,57)(48,56)(49,61)(52,59)(54,60)(55,63)(58,64);(1,3,6,18,8,19,7,20)(2,9,12,15,14,32,13,33)(4,21,23,44,24,48,29,16)(5,17,26,45,28,46,27,47)(10,34,36,30,38,42,37,56)(11,31,39,41,40,57,25,53)(22,49,51,63,52,64,54,43)(35,58,50,55,60,61,59,62);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,48)(19,44)(20,21)(25,50)(26,54)(27,51)(28,52)(31,55)(32,56)(33,42)(39,59)(40,60)(41,58)(45,64)(46,63)(47,49)(53,61)(57,62);(1,5,7,27,8,28,6,26)(2,11,13,25,14,40,12,39)(3,17,20,47,19,46,18,45)(4,22,23,51,24,52,29,54)(9,31,33,53,32,57,15,41)(10,35,36,50,38,60,37,59)(16,43,21,49,44,63,48,64)(30,55,42,61,56,62,34,58)Polycyclic presentation:
2^9|b^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319035);