Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #475185 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17029 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
659 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)(3,13)(4,9)(5,10,23,38)(6,25)(7,12)(8,16)(11,29)(14,30)(15,31,47,44)(17,32)(18,51)(19,34,22,37)(20,41)(21,36)(24,42,28,39)(26,40)(27,35)(33,49)(43,46)(45,63,64,56)(48,58,50,57)(52,59,55,62)(53,61,54,60);(1,3)(2,8,12,13)(4,18)(5,15)(6,26)(7,16)(9,33,36,51)(10,31,38,44)(11,40,25,32)(14,20)(17,29)(19,52)(21,49)(22,55)(23,47)(24,50)(27,46)(28,48)(30,41,43,35)(34,59,37,62)(39,58,42,57)(45,54)(53,64)(56,60,63,61);(1,4,7,21)(2,9,12,36)(3,14,16,46)(5,22,23,19)(6,27,29,20)(8,30,13,43)(10,37,38,34)(11,41,25,35)(15,45,47,64)(17,49,26,18)(24,53,28,54)(31,56,44,63)(32,51,40,33)(39,60,42,61)(48,52,50,55)(57,59,58,62);(1,5,6,28,7,23,29,24)(2,10,11,42,12,38,25,39)(3,15,26,48,16,47,17,50)(4,19,20,54,21,22,27,53)(8,31,40,57,13,44,32,58)(9,34,35,61,36,37,41,60)(14,45,49,55,46,64,18,52)(30,56,51,62,43,63,33,59)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=i,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,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, 382382);