Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13883 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
977 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)(3,14,18,9)(4,19)(5,24,27,11)(6,28)(7,13)(8,12)(10,22)(15,42,45,35)(16,32)(17,33,29,34)(20,31,44,40)(21,50,52,48)(23,38)(25,39,26,54)(30,37)(36,51,53,56)(41,47)(43,60)(46,59)(49,61)(55,58)(57,63)(62,64);(1,3,6,17,8,18,7,29)(2,9,28,34,12,14,13,33)(4,20,22,15,23,44,30,45)(5,16,25,46,27,47,26,55)(10,35,38,31,37,42,19,40)(11,32,54,59,24,41,39,58)(21,49,51,43,52,63,56,64)(36,60,50,57,53,62,48,61);(1,4,6,22,8,23,7,30)(2,10,13,19,12,37,28,38)(3,15,17,44,18,45,29,20)(5,21,25,51,27,52,26,56)(9,31,33,35,14,40,34,42)(11,36,39,48,24,53,54,50)(16,43,46,63,47,64,55,49)(32,57,58,60,41,61,59,62);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,32)(10,36)(12,24)(13,39)(14,41)(15,43)(17,46)(18,47)(19,48)(20,49)(22,51)(23,52)(28,54)(29,55)(30,56)(31,57)(33,58)(34,59)(35,60)(37,53)(38,50)(40,61)(42,62)(44,63)(45,64)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268100);