Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16425 |
| 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 | |
654 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)(3,14,19,34)(4,10,23,39)(5,25)(6,12)(7,30)(8,13)(9,17,33,18)(11,27)(15,35)(16,32,48,60)(20,31)(21,40,51,62)(22,37,24,38)(26,54,52,36)(28,42)(29,41)(43,50,53,47)(44,56)(45,57)(46,59)(49,58)(55,64)(61,63);(1,3,6,17,8,19,7,18)(2,9,30,34,13,33,12,14)(4,20,24,15,23,49,22,45)(5,16,27,47,29,48,28,50)(10,35,37,31,39,57,38,58)(11,32,25,53,42,60,41,43)(21,46,52,44,51,63,26,55)(36,61,40,56,54,59,62,64);(1,4,6,24,8,23,7,22)(2,10,30,37,13,39,12,38)(3,15,17,49,19,45,18,20)(5,26,27,21,29,52,28,51)(9,31,34,57,33,58,14,35)(11,40,25,54,42,62,41,36)(16,46,47,44,48,63,50,55)(32,59,53,64,60,61,43,56);(1,5,6,27,8,29,7,28)(2,11,12,41,13,42,30,25)(3,16,17,47,19,48,18,50)(4,21,22,26,23,51,24,52)(9,32,14,43,33,60,34,53)(10,36,37,40,39,54,38,62)(15,44,20,46,45,55,49,63)(31,56,57,59,58,64,35,61)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369484);