Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1067 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)(3,15,18,9)(4,21,25,40)(5,26,29,38)(6,13,7,12)(10,30,37,24)(11,28,39,27)(16,35,44,59)(17,46,48,56)(19,33,20,34)(22,31,45,41)(23,36,55,62)(32,50,42,49)(43,58,47,60)(51,53,57,54)(52,64,63,61);(1,3,6,20,8,18,7,19)(2,9,12,34,14,15,13,33)(4,22,24,44,25,45,30,16)(5,17,27,50,29,48,28,49)(10,35,21,41,37,59,40,31)(11,32,26,56,39,42,38,46)(23,52,53,47,55,63,54,43)(36,61,51,60,62,64,57,58);(1,4)(2,10)(3,16)(5,23)(6,30)(7,24)(8,25)(9,31)(11,36)(12,40)(13,21)(14,37)(15,41)(17,47)(18,44)(19,22)(20,45)(26,57)(27,54)(28,53)(29,55)(32,60)(33,35)(34,59)(38,51)(39,62)(42,58)(43,48)(46,64)(49,63)(50,52)(56,61);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,29)(9,32)(10,36)(12,26)(13,38)(14,39)(15,42)(16,43)(18,48)(19,49)(20,50)(21,51)(22,52)(24,53)(25,55)(30,54)(31,58)(33,46)(34,56)(35,61)(37,62)(40,57)(41,60)(44,47)(45,63)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,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, 362883);