Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15701 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,17,9,18,40,29,34)(4,19,22,37,23,10,30,47)(5,24,31,54,27,11,26,39)(15,42,49,32,20,41,44,35)(16,38,25,33,45,53,56,59)(21,36,51,64,52,46,57,62)(43,61,50,58,48,63,55,60);(1,3,8,17,7,18,6,29)(2,9,12,40,13,34,28,14)(4,20,22,44,23,15,30,49)(5,25,31,45,27,56,26,16)(10,35,47,42,19,32,37,41)(11,38,39,33,24,53,54,59)(21,50,51,48,52,55,57,43)(36,61,64,58,46,63,62,60);(1,4,6,22,7,23,8,30)(2,10,28,47,13,19,12,37)(3,15,17,44,18,20,29,49)(5,21,26,51,27,52,31,57)(9,32,40,42,34,35,14,41)(11,36,54,64,24,46,39,62)(16,43,25,55,45,48,56,50)(33,58,53,61,59,60,38,63);(1,5,6,26,7,27,8,31)(2,11,12,39,13,24,28,54)(3,16,17,25,18,45,29,56)(4,21,22,51,23,52,30,57)(9,33,14,38,34,59,40,53)(10,36,37,62,19,46,47,64)(15,43,44,55,20,48,49,50)(32,58,41,63,35,60,42,61)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349325);