Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470221 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16679 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,30,34,18,33,17,9)(4,19,21,10,23,51,22,37)(5,24,28,11,27,39,26,40)(15,31,44,50,45,35,54,41)(16,32,48,42,49,59,47,55)(20,36,25,56,53,61,52,38)(43,62,46,60,64,58,63,57);(1,3,6,17,8,18,7,30)(2,9,12,14,13,34,29,33)(4,15,21,44,23,45,22,54)(5,16,26,47,27,49,28,48)(10,35,19,50,37,31,51,41)(11,32,39,42,40,59,24,55)(20,43,25,46,53,64,52,63)(36,60,38,62,61,57,56,58);(1,4)(2,10,13,37)(3,15,18,45)(5,25,27,52)(6,21)(7,22)(8,23)(9,31)(11,38)(12,19,29,51)(14,41)(16,46)(17,44,30,54)(20,26,53,28)(24,36)(32,58,59,62)(33,50)(34,35)(39,61)(40,56)(42,60,55,57)(43,48)(47,64)(49,63);(1,5,7,28,8,27,6,26)(2,11,12,39,13,40,29,24)(3,16,30,48,18,49,17,47)(4,20,21,25,23,53,22,52)(9,32,14,42,34,59,33,55)(10,36,51,56,37,61,19,38)(15,43,44,46,45,64,54,63)(31,57,50,62,35,60,41,58)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377417);