Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14306 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,19,9,20,33,18,34)(4,21,24,42,25,10,30,37)(5,11,29,41,28,40,27,39)(16,35,45,31,46,51,22,53)(17,43,49,32,50,61,48,62)(23,52,26,57,55,36,56,38)(44,63,47,59,58,60,54,64);(1,3,8,18,7,20,6,19)(2,9,14,15,13,34,12,33)(4,22,30,16,25,45,24,46)(5,17,29,48,28,50,27,49)(10,35,42,31,21,51,37,53)(11,32,41,43,40,62,39,61)(23,54,56,44,55,47,26,58)(36,63,57,59,52,60,38,64);(1,4)(2,10)(3,16)(5,26)(6,30)(7,25)(8,24)(9,31)(11,38)(12,42)(13,21)(14,37)(15,35)(17,47)(18,22)(19,45)(20,46)(23,27)(28,56)(29,55)(32,60)(33,51)(34,53)(36,39)(40,57)(41,52)(43,59)(44,48)(49,58)(50,54)(61,64)(62,63);(1,5)(2,11)(3,17)(4,23,25,55)(6,27)(7,28)(8,29)(9,32)(10,36,21,52)(12,39)(13,40)(14,41)(15,43)(16,44,46,58)(18,48)(19,49)(20,50)(22,54,45,47)(24,26,30,56)(31,59,53,64)(33,61)(34,62)(35,63,51,60)(37,38,42,57)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288894);