Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16340 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,19,33,18,32,17,9)(4,10,29,36,23,37,22,42)(5,11,27,39,26,41,25,40)(15,34,20,30,46,57,45,58)(16,47,51,60,50,43,49,31)(21,52,54,61,53,35,24,38)(44,59,63,62,55,64,48,56);(1,3,7,18)(2,9)(4,20,23,45)(5,16,26,50)(6,17,8,19)(10,34)(11,31)(12,14)(13,33)(15,29,46,22)(21,48,53,63)(24,55,54,44)(25,49,27,51)(28,32)(30,36)(35,59)(37,57)(38,62)(39,47)(40,43)(41,60)(42,58)(52,64)(56,61);(1,4)(2,10)(3,15)(5,24)(6,29)(7,23)(8,22)(9,30)(11,38)(12,42)(13,37)(14,34)(16,48)(17,20)(18,46)(19,45)(21,25)(26,54)(27,53)(28,36)(31,59)(32,57)(33,58)(35,39)(40,52)(41,61)(43,62)(44,49)(47,56)(50,63)(51,55)(60,64);(1,5)(2,11,13,41)(3,16)(4,21)(6,27)(7,26)(8,25)(9,31,33,60)(10,35,37,52)(12,40,28,39)(14,43,32,47)(15,44)(17,51)(18,50)(19,49)(20,48)(22,54)(23,53)(24,29)(30,56,58,62)(34,59,57,64)(36,61,42,38)(45,63)(46,55)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366645);