Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,27)(3,14,17,9,18,34,28,33)(4,19,21,36,22,10,29,37)(5,23,25,11,26,39,30,40)(15,41,45,31,46,60,44,50)(16,47,49,42,48,32,24,38)(20,52,56,51,55,57,54,35)(43,63,53,61,64,59,58,62);(1,3)(2,9)(4,15)(5,24)(6,28)(7,18)(8,17)(10,31)(11,38)(12,34)(13,33)(14,27)(16,25)(19,50)(20,53)(21,45)(22,46)(23,32)(26,49)(29,44)(30,48)(35,61)(36,41)(37,60)(39,47)(40,42)(43,54)(51,62)(52,59)(55,58)(56,64)(57,63);(1,4,8,21,7,22,6,29)(2,10,27,36,13,19,12,37)(3,15,28,44,18,46,17,45)(5,20,25,56,26,55,30,54)(9,31,34,60,33,50,14,41)(11,35,23,57,40,51,39,52)(16,43,24,58,48,64,49,53)(32,59,38,62,47,63,42,61);(1,5)(2,11)(3,16)(4,20)(6,30)(7,26)(8,25)(9,32)(10,35)(12,39)(13,40)(14,42)(15,43)(17,49)(18,48)(19,51)(21,56)(22,55)(23,27)(24,28)(29,54)(31,59)(33,47)(34,38)(36,57)(37,52)(41,61)(44,58)(45,53)(46,64)(50,63)(60,62)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293114);