Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,25,37,26,42,24,10)(5,11,29,39,30,41,28,40)(16,31,46,43,47,60,22,35)(17,44,27,57,49,38,48,32)(23,52,55,62,56,50,54,36)(45,64,51,63,53,59,58,61);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,35)(11,38)(12,33)(13,15)(14,34)(16,25)(17,28)(21,31)(23,53)(24,47)(26,46)(29,49)(30,48)(32,39)(36,59)(37,43)(40,57)(41,44)(42,60)(45,56)(50,63)(51,54)(52,61)(55,58)(62,64);(1,4)(2,10,14,37)(3,16)(5,23)(6,25)(7,24)(8,26)(9,31,34,60)(11,36,41,62)(12,42,13,21)(15,43,33,35)(17,45)(18,22)(19,46)(20,47)(27,58)(28,55)(29,54)(30,56)(32,59,57,64)(38,61,44,63)(39,50,40,52)(48,51)(49,53);(1,5,8,30)(2,11,14,41)(3,17,20,49)(4,23,26,56)(6,29,7,28)(9,32,34,57)(10,36,37,62)(12,40,13,39)(15,44,33,38)(16,45,47,53)(18,48,19,27)(21,50,42,52)(22,51,46,58)(24,55,25,54)(31,59,60,64)(35,61,43,63)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 293111);