Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384166 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14509 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,20,9,19,34,18,35)(4,21,24,41,25,10,31,37)(5,26,30,40,29,39,28,11)(16,42,44,49,22,51,48,32)(17,45,47,38,46,33,27,58)(23,53,55,50,56,57,60,36)(43,63,59,64,52,61,54,62);(1,3)(2,9)(4,22)(5,27)(6,20)(7,19)(8,18)(10,32)(11,38)(12,34)(13,35)(14,15)(16,25)(17,28)(21,49)(23,54)(24,44)(26,45)(29,47)(30,46)(31,48)(33,39)(36,62)(37,51)(40,58)(41,42)(43,60)(50,64)(52,55)(53,61)(56,59)(57,63);(1,4,8,31,7,25,6,24)(2,10,14,41,13,21,12,37)(3,16,18,44,19,22,20,48)(5,23,30,60,29,56,28,55)(9,32,15,42,35,49,34,51)(11,36,26,57,40,50,39,53)(17,43,27,59,46,52,47,54)(33,61,38,62,45,63,58,64);(1,5,8,30,7,29,6,28)(2,11,12,39,13,40,14,26)(3,17,20,47,19,46,18,27)(4,23,31,60,25,56,24,55)(9,33,15,38,35,45,34,58)(10,36,37,53,21,50,41,57)(16,43,48,54,22,52,44,59)(32,61,42,62,49,63,51,64)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295223);