Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14795 |
| 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 | |
1333 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)(3,15,19,33)(4,10,24,37)(5,25,28,11)(6,13,7,12)(9,20,32,18)(16,42,44,34)(17,45,47,41)(21,50,49,30)(22,52,54,35)(23,40,29,36)(26,38,27,39)(31,46,55,48)(43,58,62,61)(51,57,63,64)(53,59,56,60);(1,3)(2,9)(4,21)(5,17)(6,20)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,23)(22,51)(24,49)(25,55)(26,48)(27,46)(28,47)(29,44)(30,36)(35,58)(37,42)(38,45)(39,41)(40,50)(43,53)(52,61)(54,63)(56,62)(57,59)(60,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,43)(18,21)(19,44)(20,49)(25,52)(26,53)(27,56)(28,54)(31,57)(32,50)(33,42)(38,59)(39,60)(41,58)(45,61)(46,51)(47,62)(48,63)(55,64);(1,5,6,26,8,28,7,27)(2,11,12,38,14,25,13,39)(3,17,20,48,19,47,18,46)(4,22,23,53,24,54,29,56)(9,31,33,45,32,55,15,41)(10,35,36,59,37,52,40,60)(16,43,49,63,44,62,21,51)(30,57,42,61,50,64,34,58)Polycyclic presentation:
2^9|a^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307246);