Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14260 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
933 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)(3,14,18,34)(4,19,24,10)(5,11)(6,12)(7,29)(8,13)(9,17,33,30)(15,43)(16,42,48,61)(20,51)(21,50,53,36)(22,41,23,37)(25,54,52,38)(26,39)(27,56)(28,40)(31,49)(32,47,60,57)(35,45)(44,59)(46,63)(55,62)(58,64);(1,3)(2,9,13,33)(4,20)(5,16)(6,17)(7,30)(8,18)(10,35,19,43)(11,32,40,60)(12,14,29,34)(15,23)(21,46)(22,45)(24,49)(25,55)(26,47)(27,57)(28,48)(31,41,51,37)(36,62,50,59)(38,58,54,63)(39,42,56,61)(44,52)(53,64);(1,4)(2,10)(3,15)(5,25)(6,23)(7,22)(8,24)(9,31)(11,38)(12,41)(13,19)(14,35)(16,46)(17,49)(18,45)(20,30)(21,26)(27,53)(28,52)(29,37)(32,59)(33,51)(34,43)(36,39)(40,54)(42,58)(44,47)(48,64)(50,56)(55,57)(60,62)(61,63);(1,5,6,26,8,28,7,27)(2,11,12,39,13,40,29,56)(3,16,17,47,18,48,30,57)(4,21,22,25,24,53,23,52)(9,32,14,42,33,60,34,61)(10,36,37,38,19,50,41,54)(15,44,20,46,45,55,49,64)(31,58,43,59,51,63,35,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287384);