Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458571 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
959 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)(3,14)(4,10)(5,23,26,11)(6,27)(7,12)(8,13)(9,17)(15,35)(16,43,49,60)(18,33)(19,51)(20,52,53,36)(21,38)(22,39)(24,40,54,56)(25,41,28,42)(29,34)(30,37)(31,45)(32,48,55,50)(44,62,57,59)(46,61)(47,64,63,58);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,27,33)(4,19,21,15,22,45,30,46)(5,16,28,50,26,49,25,48)(10,35,37,51,39,61,38,31)(11,32,42,43,23,55,41,60)(20,47,54,44,53,63,24,57)(36,62,56,64,52,59,40,58);(1,4)(2,10)(3,15)(5,24,26,54)(6,21)(7,30)(8,22)(9,31)(11,40,23,56)(12,37)(13,39)(14,35)(16,47,49,63)(17,45)(18,46)(19,29)(20,25,53,28)(27,38)(32,59,55,62)(33,61)(34,51)(36,41,52,42)(43,58,60,64)(44,48,57,50);(1,5)(2,11)(3,16)(4,20)(6,28)(7,25)(8,26)(9,32)(10,36)(12,42)(13,23)(14,43)(15,44)(17,50)(18,49)(19,47)(21,54)(22,53)(24,30)(27,41)(29,48)(31,58)(33,60)(34,55)(35,62)(37,56)(38,40)(39,52)(45,63)(46,57)(51,64)(59,61)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365767);