Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388397 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1232 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)(3,14)(4,10,23,38)(5,24,28,40)(6,29)(7,12)(8,13)(9,17)(11,26,41,27)(15,44,46,34)(16,32,47,55)(18,42)(19,33)(20,31,49,57)(21,51)(22,37,30,36)(25,43,48,39)(35,52)(45,59)(50,62)(53,60)(54,61)(56,64)(58,63);(1,3)(2,9)(4,20)(5,25)(6,19)(7,17)(8,18)(10,34)(11,39)(12,42)(13,33)(14,29)(15,22)(16,26)(21,45)(23,49)(24,55)(27,47)(28,48)(30,46)(31,37)(32,40)(35,59)(36,57)(38,44)(41,43)(50,52)(51,58)(53,64)(54,63)(56,61)(60,62);(1,4,6,30,8,23,7,22)(2,10,12,37,13,38,29,36)(3,15,17,49,18,46,19,20)(5,21,27,52,28,53,26,54)(9,31,14,34,33,57,42,44)(11,35,24,51,41,61,40,60)(16,45,48,63,47,64,25,50)(32,56,43,58,55,59,39,62);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,32)(10,35)(12,24)(13,41)(14,43)(15,45)(17,48)(18,47)(19,25)(20,50)(22,54)(23,53)(29,40)(30,52)(31,56)(33,55)(34,58)(36,60)(37,51)(38,61)(39,42)(44,62)(46,64)(49,63)(57,59)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295593);