Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394285 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2544 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)(5,11)(6,26)(7,12)(8,13)(9,17)(15,33)(16,40)(18,31)(19,46)(20,48)(21,36)(22,37)(23,51)(24,39)(25,38)(27,32)(28,35)(29,42)(30,44)(34,50)(41,61)(43,57)(45,56)(47,54)(49,59)(52,55)(53,60)(58,63)(62,64);(1,3,6,17,8,18,7,27)(2,9,12,14,13,32,26,31)(4,19,21,15,22,42,28,43)(5,16,23,44,24,45,25,52)(10,33,35,46,37,57,36,29)(11,30,38,40,39,55,51,56)(20,47,49,41,50,62,53,63)(34,58,59,64,48,61,60,54);(1,4)(2,10)(3,15)(5,20)(6,21)(7,28)(8,22)(9,29)(11,34)(12,35)(13,37)(14,33)(16,41)(17,42)(18,43)(19,27)(23,49)(24,50)(25,53)(26,36)(30,54)(31,57)(32,46)(38,59)(39,48)(40,58)(44,62)(45,63)(47,52)(51,60)(55,64)(56,61);(1,5)(2,11)(3,16)(4,20)(6,23)(7,25)(8,24)(9,30)(10,34)(12,38)(13,39)(14,40)(15,41)(17,44)(18,45)(19,47)(21,49)(22,50)(26,51)(27,52)(28,53)(29,54)(31,56)(32,55)(33,58)(35,59)(36,60)(37,48)(42,62)(43,63)(46,64)(57,61)Polycyclic presentation:
2^9|b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301481);