Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14134 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,25,37,26,40,24,10)(5,27,28,11,30,38,29,39)(16,31,43,55,44,56,22,35)(17,32,46,57,47,58,45,41)(23,36,50,48,52,60,51,53)(42,61,62,64,63,59,49,54);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,31)(23,49)(24,44)(26,43)(27,41)(28,46)(29,45)(30,47)(36,59)(37,55)(38,57)(39,58)(40,56)(42,50)(48,54)(51,63)(52,62)(53,64)(60,61);(1,4)(2,10,14,37)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31,34,56)(11,36,39,60)(12,21,13,40)(15,35,33,55)(17,42)(18,43)(19,22)(20,44)(27,53,38,48)(28,50)(29,51)(30,52)(32,54,58,64)(41,59,57,61)(45,49)(46,62)(47,63);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,38)(13,27)(14,39)(15,41)(16,42)(18,46)(19,45)(20,47)(21,48)(22,49)(24,51)(25,50)(26,52)(31,54)(33,57)(34,58)(35,59)(37,60)(40,53)(43,62)(44,63)(55,61)(56,64)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,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, 281326);