Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #445016 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
728 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)(3,15,19,34,18,9,20,40)(4,21,26,36,25,41,24,10)(5,27,31,11,30,38,29,39)(16,43,45,42,22,51,49,32)(17,46,47,37,28,56,48,33)(23,35,55,59,54,60,53,50)(44,57,62,64,52,58,63,61);(1,3,6,20,7,18,8,19)(2,9,12,40,14,15,13,34)(4,22,24,45,25,16,26,49)(5,28,29,47,30,17,31,48)(10,32,21,43,36,42,41,51)(11,37,38,56,39,33,27,46)(23,44,53,63,54,52,55,62)(35,58,59,61,60,57,50,64);(1,4,7,25)(2,10)(3,16,18,22)(5,23,30,54)(6,24,8,26)(9,32)(11,35)(12,21)(13,41)(14,36)(15,42)(17,44,28,52)(19,45,20,49)(27,50)(29,53,31,55)(33,57)(34,51)(37,58)(38,59)(39,60)(40,43)(46,64)(47,62,48,63)(56,61);(1,5,7,30)(2,11,14,39)(3,17,18,28)(4,23,25,54)(6,31,8,29)(9,33,15,37)(10,35,36,60)(12,27,13,38)(16,44,22,52)(19,48,20,47)(21,50,41,59)(24,55,26,53)(32,57,42,58)(34,46,40,56)(43,61,51,64)(45,63,49,62)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=g,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 352213);