Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14023 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1634 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)(3,14,17,9,18,34,30,33)(4,19,24,38,23,10,22,37)(5,25,26,11,28,39,27,40)(15,41,44,31,45,35,20,50)(16,32,48,58,47,59,46,42)(21,52,53,49,54,56,55,36)(43,57,62,60,63,64,51,61);(1,3,7,17,8,18,6,30)(2,9,12,34,13,33,29,14)(4,20,24,15,23,44,22,45)(5,16,26,48,28,47,27,46)(10,35,37,50,19,41,38,31)(11,32,39,58,40,59,25,42)(21,51,53,43,54,62,55,63)(36,60,52,64,49,61,56,57);(1,4,6,22,8,23,7,24)(2,10,12,37,13,19,29,38)(3,15,30,20,18,45,17,44)(5,21,27,55,28,54,26,53)(9,31,34,35,33,50,14,41)(11,36,39,52,40,49,25,56)(16,43,46,51,47,63,48,62)(32,57,58,60,59,64,42,61);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,32)(10,36)(12,39)(13,40)(14,42)(15,43)(17,48)(18,47)(19,49)(20,51)(22,55)(23,54)(24,53)(25,29)(30,46)(31,57)(33,59)(34,58)(35,60)(37,52)(38,56)(41,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276066);