Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453917 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16222 |
| 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 | |
627 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)(3,13)(4,18)(5,10,23,34)(6,25)(7,12)(8,16)(9,21)(11,27)(14,39)(15,29,44,37)(17,30)(19,48)(20,51,52,46)(22,47)(24,36,26,35)(28,50)(31,41)(32,54,60,53)(33,55)(38,42)(40,61,49,64)(43,58,45,57)(56,63,59,62);(1,3)(2,8,12,13)(4,19,22,14)(5,15,23,44)(6,17)(7,16)(9,31)(10,29)(11,30,25,38)(18,39)(20,49)(21,50,55,41)(24,43,26,45)(27,42)(28,33)(32,59,60,56)(34,37)(35,57)(36,58)(40,52)(46,61,51,64)(47,48)(53,63)(54,62);(1,4,6,21,7,22,27,55)(2,9,11,18,12,33,25,47)(3,14,17,41,16,19,42,50)(5,20,26,54,23,52,24,53)(8,28,30,48,13,31,38,39)(10,32,36,46,34,60,35,51)(15,40,45,63,44,49,43,62)(29,56,58,64,37,59,57,61);(1,5,6,26,7,23,27,24)(2,10,11,36,12,34,25,35)(3,15,42,43,16,44,17,45)(4,20,21,54,22,52,55,53)(8,29,38,57,13,37,30,58)(9,32,18,46,33,60,47,51)(14,40,50,62,19,49,41,63)(28,56,39,61,31,59,48,64)Polycyclic presentation:
2^9|a^2=eg,b^2=gh,c^2=efg,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,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, 361113);