Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,30)(3,14,17,34,18,33,28,9)(4,19,21,37,22,10,29,36)(5,23,25,39,26,53,27,11)(15,40,42,31,43,58,44,46)(16,32,24,38,45,59,55,60)(20,48,50,47,51,52,56,35)(41,57,49,61,63,64,54,62);(1,3,8,18)(2,9,13,34)(4,15,22,43)(5,24,26,55)(6,28,7,17)(10,31,19,46)(11,38,39,60)(12,33,30,14)(16,27,45,25)(20,49,51,54)(21,44,29,42)(23,32,53,59)(35,61,47,62)(36,40,37,58)(41,56,63,50)(48,57,52,64);(1,4)(2,10)(3,15)(5,20)(6,29)(7,21)(8,22)(9,31)(11,35)(12,37)(13,19)(14,40)(16,41)(17,44)(18,43)(23,52)(24,54)(25,56)(26,51)(27,50)(28,42)(30,36)(32,57)(33,58)(34,46)(38,62)(39,47)(45,63)(48,53)(49,55)(59,64)(60,61);(1,5)(2,11,13,39)(3,16,18,45)(4,20)(6,25)(7,27)(8,26)(9,32)(10,35,19,47)(12,23,30,53)(14,38)(15,41,43,63)(17,24,28,55)(21,50)(22,51)(29,56)(31,57)(33,60)(34,59)(36,48,37,52)(40,62)(42,49,44,54)(46,64)(58,61)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334625);