Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14896 |
| 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 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,18,9)(4,20,24,40)(5,25,28,11)(6,13,7,12)(10,30,36,23)(16,34,44,57)(17,32,46,42)(19,33,29,39)(21,31,45,41)(22,50,52,48)(26,37,27,38)(35,51,53,55)(43,62,63,59)(47,60,54,58)(49,61,64,56);(1,3,6,29,8,18,7,19)(2,9,12,39,14,15,13,33)(4,21,23,44,24,45,30,16)(5,17,26,54,28,46,27,47)(10,34,20,41,36,57,40,31)(11,32,37,60,25,42,38,58)(22,49,51,63,52,64,55,43)(35,59,48,61,53,62,50,56);(1,4)(2,10)(3,16,18,44)(5,22)(6,30)(7,23)(8,24)(9,31,15,41)(11,35)(12,40)(13,20)(14,36)(17,43,46,63)(19,21,29,45)(25,53)(26,55)(27,51)(28,52)(32,56,42,61)(33,34,39,57)(37,50)(38,48)(47,49,54,64)(58,59,60,62);(1,5,7,27,8,28,6,26)(2,11,13,38,14,25,12,37)(3,17,19,47,18,46,29,54)(4,22,23,51,24,52,30,55)(9,32,33,58,15,42,39,60)(10,35,20,48,36,53,40,50)(16,43,21,49,44,63,45,64)(31,56,34,59,41,61,57,62)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=e,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311827);