Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15508 |
| 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 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,6,11,20,34)(3,13,28,32,16,8,17,39)(4,18,22,36,23,40,29,9)(5,24,49,38,26,10,27,54)(14,41,45,33,19,46,43,30)(15,31,44,58,25,37,55,60)(21,50,56,53,52,47,51,35)(42,62,63,59,48,61,64,57);(1,3,6,16)(2,8,11,13)(4,19)(5,25,26,15)(7,28,20,17)(9,33)(10,37,24,31)(12,39,34,32)(14,23)(18,46)(21,48)(22,43)(27,44,49,55)(29,45)(30,36)(35,59)(38,58,54,60)(40,41)(42,52)(47,62)(50,61)(51,63)(53,57)(56,64);(1,4,20,22,6,23,7,29)(2,9,34,18,11,36,12,40)(3,14,28,43,16,19,17,45)(5,21,49,51,26,52,27,56)(8,30,39,46,13,33,32,41)(10,35,54,47,24,53,38,50)(15,42,55,63,25,48,44,64)(31,57,60,62,37,59,58,61);(1,5,6,26)(2,10,11,24)(3,15,16,25)(4,21,23,52)(7,27,20,49)(8,31,13,37)(9,35,36,53)(12,38,34,54)(14,42,19,48)(17,44,28,55)(18,47,40,50)(22,51,29,56)(30,57,33,59)(32,58,39,60)(41,61,46,62)(43,63,45,64)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342027);