Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #443903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15768 |
| 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 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,28)(3,14,17,32,18,33,27,9)(4,10,20,36,21,37,22,54)(5,23,25,11,26,39,29,40)(15,42,44,30,45,41,53,34)(16,46,24,56,48,31,47,38)(19,49,51,35,52,60,55,61)(43,57,50,62,63,58,64,59);(1,3,8,17,7,18,6,27)(2,9,28,14,13,32,12,33)(4,15,20,53,21,45,22,44)(5,24,29,48,26,47,25,16)(10,34,36,41,37,30,54,42)(11,38,23,46,40,56,39,31)(19,50,51,43,52,64,55,63)(35,59,60,58,61,62,49,57);(1,4,8,22,7,21,6,20)(2,10,12,36,13,37,28,54)(3,15,17,44,18,45,27,53)(5,19,29,55,26,52,25,51)(9,30,33,42,32,34,14,41)(11,35,39,60,40,61,23,49)(16,43,24,50,48,63,47,64)(31,57,56,59,46,58,38,62);(1,5,6,25,7,26,8,29)(2,11,12,39,13,40,28,23)(3,16,17,24,18,48,27,47)(4,19,20,51,21,52,22,55)(9,31,14,38,32,46,33,56)(10,35,36,60,37,61,54,49)(15,43,44,50,45,63,53,64)(30,57,41,62,34,58,42,59)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=eg,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351099);