Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1107 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)(3,15,18,9)(4,21,25,39)(5,26,28,40)(6,13,7,12)(10,29,37,24)(11,30,38,27)(16,42,22,52)(17,45,47,56)(19,33,20,34)(23,36,54,60)(31,44,35,49)(32,48,41,50)(43,59,46,58)(51,53,57,55)(61,62,64,63);(1,3,6,19,8,18,7,20)(2,9,12,33,14,15,13,34)(4,22,24,49,25,16,29,44)(5,17,27,48,28,47,30,50)(10,35,21,42,37,31,39,52)(11,32,26,56,38,41,40,45)(23,46,53,63,54,43,55,62)(36,59,51,61,60,58,57,64);(1,4)(2,10)(3,16)(5,23)(6,29)(7,24)(8,25)(9,31)(11,36)(12,39)(13,21)(14,37)(15,35)(17,46)(18,22)(19,49)(20,44)(26,57)(27,55)(28,54)(30,53)(32,59)(33,42)(34,52)(38,60)(40,51)(41,58)(43,47)(45,61)(48,62)(50,63)(56,64);(1,5)(2,11)(3,17)(4,23)(6,27)(7,30)(8,28)(9,32)(10,36)(12,26)(13,40)(14,38)(15,41)(16,43)(18,47)(19,48)(20,50)(21,51)(22,46)(24,53)(25,54)(29,55)(31,58)(33,56)(34,45)(35,59)(37,60)(39,57)(42,61)(44,62)(49,63)(52,64)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362765);