Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15926 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,19,42,31,34)(4,20,23,37,25,10,24,38)(5,26,30,11,29,40,28,41)(16,35,50,60,44,47,21,32)(17,33,27,58,46,56,45,39)(22,51,53,36,55,57,54,48)(43,59,63,61,52,62,49,64);(1,3,8,18,7,19,6,31)(2,9,12,42,14,34,13,15)(4,21,23,16,25,50,24,44)(5,27,30,46,29,45,28,17)(10,35,38,60,20,47,37,32)(11,39,40,33,41,58,26,56)(22,52,53,49,55,43,54,63)(36,62,57,64,48,59,51,61);(1,4,8,24,7,25,6,23)(2,10,13,38,14,20,12,37)(3,16,31,50,19,44,18,21)(5,22,30,54,29,55,28,53)(9,32,42,47,34,60,15,35)(11,36,26,57,41,48,40,51)(17,43,45,63,46,52,27,49)(33,59,58,64,56,62,39,61);(1,5)(2,11)(3,17,19,46)(4,22)(6,28)(7,29)(8,30)(9,33,34,56)(10,36)(12,40)(13,26)(14,41)(15,39,42,58)(16,43,44,52)(18,27,31,45)(20,48)(21,49,50,63)(23,53)(24,54)(25,55)(32,59,60,62)(35,61,47,64)(37,51)(38,57)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=ei,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356130);