Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #399150 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14776 |
| 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 | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,19,33)(4,10,24,37)(5,25,27,40)(6,14,8,12)(9,20,32,18)(11,29,38,26)(16,34,44,55)(17,45,47,31)(21,30,43,56)(22,50,53,59)(23,39,28,36)(35,51,60,52)(41,48,57,46)(42,61,63,54)(49,58,62,64);(1,3,8,20,7,19,6,18)(2,9,14,33,13,32,12,15)(4,21,23,16,24,43,28,44)(5,17,29,48,27,47,26,46)(10,34,36,30,37,55,39,56)(11,31,40,57,38,45,25,41)(22,49,51,42,53,62,52,63)(35,58,59,54,60,64,50,61);(1,4,8,23,7,24,6,28)(2,10,14,36,13,37,12,39)(3,16,20,43,19,44,18,21)(5,22,29,51,27,53,26,52)(9,30,33,55,32,56,15,34)(11,35,40,59,38,60,25,50)(17,42,48,62,47,63,46,49)(31,54,57,64,45,61,41,58);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,29)(9,31)(10,35)(12,25)(13,38)(14,40)(15,41)(16,42)(18,46)(19,47)(20,48)(21,49)(23,51)(24,53)(28,52)(30,54)(32,45)(33,57)(34,58)(36,59)(37,60)(39,50)(43,62)(44,63)(55,64)(56,61)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=ef,e^2=i,g^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,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306347);