Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14019 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 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)(4,21,26,40)(5,27)(6,13)(7,12)(8,14)(9,18)(10,25,36,24)(11,28)(16,31,44,56)(17,32)(19,33)(20,39)(22,34,43,55)(23,35,51,60)(29,38)(30,37)(41,45)(42,61,63,59)(46,58)(47,57)(48,52,53,50)(49,54,62,64);(1,3,6,19,8,20,7,18)(2,9,13,15,14,33,12,39)(4,22,24,44,26,43,25,16)(5,17,29,45,30,46,28,47)(10,34,40,56,36,55,21,31)(11,32,27,41,38,58,37,57)(23,49,52,63,51,62,50,42)(35,59,53,64,60,61,48,54);(1,4,7,25,8,26,6,24)(2,10,12,21,14,36,13,40)(3,16,19,22,20,44,18,43)(5,23,28,50,30,51,29,52)(9,31,15,34,33,56,39,55)(11,35,37,48,38,60,27,53)(17,42,45,49,46,63,47,62)(32,54,41,59,58,64,57,61);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,35)(12,37)(13,27)(14,38)(15,41)(16,42)(18,47)(19,45)(20,46)(21,48)(22,49)(24,52)(25,50)(26,51)(31,54)(33,58)(34,59)(36,60)(39,57)(40,53)(43,62)(44,63)(55,61)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275708);