Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13869 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,34,20,9,19,33)(4,21,24,37,26,38,25,10)(5,27,28,40,29,11,30,39)(16,43,45,41,46,35,22,31)(17,32,48,59,49,42,47,58)(23,52,54,56,55,50,53,36)(44,60,51,61,63,62,64,57);(1,3)(2,9)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35)(11,32)(12,15)(13,33)(14,34)(16,25)(21,41)(23,51)(24,46)(26,45)(27,42)(28,47)(29,49)(30,48)(31,38)(36,60)(37,43)(39,58)(40,59)(44,54)(50,61)(52,57)(53,63)(55,64)(56,62);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31,15,41)(11,36,27,56)(13,21,14,38)(17,44)(18,22)(19,45)(20,46)(28,53)(29,55)(30,54)(32,57,42,61)(33,35,34,43)(39,50,40,52)(47,51)(48,64)(49,63)(58,60,59,62);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,39)(14,40)(15,42)(16,44)(18,47)(19,48)(20,49)(21,50)(22,51)(24,53)(25,54)(26,55)(31,57)(33,58)(34,59)(35,60)(37,56)(38,52)(41,61)(43,62)(45,64)(46,63)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267098);