Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423245 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15313 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1468 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)(6,28)(7,13)(8,12)(9,17)(15,44)(16,42)(18,32)(19,43)(20,30)(21,51)(22,36)(23,37)(24,55)(25,41)(26,39)(27,40)(29,35)(31,48)(33,46)(34,53)(38,47)(45,58)(49,57)(50,59)(52,60)(54,61)(56,63)(62,64);(1,3,6,17,7,19,8,18)(2,9,12,14,13,32,28,43)(4,20,22,46,23,49,29,15)(5,24,25,47,27,48,26,16)(10,33,35,30,37,44,36,57)(11,38,39,55,40,42,41,31)(21,45,52,63,53,64,54,50)(34,59,60,62,51,56,61,58);(1,4,8,22,7,23,6,29)(2,10,28,35,13,37,12,36)(3,15,17,49,19,46,18,20)(5,21,25,54,27,53,26,52)(9,30,14,33,32,57,43,44)(11,34,39,61,40,51,41,60)(16,45,48,63,47,64,24,50)(31,56,42,58,55,59,38,62);(1,5,7,27)(2,11,13,40)(3,16,19,47)(4,21,23,53)(6,26,8,25)(9,31,32,55)(10,34,37,51)(12,41,28,39)(14,42,43,38)(15,45,46,64)(17,48,18,24)(20,50,49,63)(22,54,29,52)(30,56,57,59)(33,58,44,62)(35,61,36,60)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330442);