Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16557 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,8,12)(3,15,18,9,19,32,20,33)(4,10,23,37,24,38,25,36)(5,11,29,40,28,41,27,42)(16,34,45,58,46,57,21,30)(17,47,51,60,50,43,49,31)(22,52,54,61,53,35,26,39)(44,59,63,62,55,64,48,56);(1,3,8,20,7,19,6,18)(2,9,13,32,14,33,12,15)(4,21,25,46,24,45,23,16)(5,17,27,49,28,50,29,51)(10,34,37,58,38,57,36,30)(11,31,40,47,41,60,42,43)(22,48,26,55,53,63,54,44)(35,59,39,62,52,64,61,56);(1,4,7,24)(2,10,14,38)(3,16,19,46)(5,26,28,54)(6,25,8,23)(9,30,33,58)(11,39,41,61)(12,37,13,36)(15,34,32,57)(17,48,50,63)(18,21,20,45)(22,29,53,27)(31,59,60,64)(35,42,52,40)(43,62,47,56)(44,51,55,49);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,27)(9,31)(10,35)(12,42)(13,40)(14,41)(15,43)(16,44)(18,51)(19,50)(20,49)(21,48)(23,54)(24,53)(25,26)(30,56)(32,47)(33,60)(34,59)(36,61)(37,39)(38,52)(45,63)(46,55)(57,64)(58,62)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374245);