Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,9,18,32,19,33,17,31)(4,20,25,10,23,37,24,36)(5,11,26,38,28,39,27,40)(15,41,44,29,43,34,21,49)(16,30,45,56,47,57,46,58)(22,51,53,60,52,48,54,35)(42,59,63,64,62,61,50,55);(1,3,7,18,8,19,6,17)(2,9,13,32,14,33,12,31)(4,21,25,15,23,44,24,43)(5,16,26,45,28,47,27,46)(10,34,37,49,36,41,20,29)(11,30,38,56,39,57,40,58)(22,50,53,42,52,63,54,62)(35,59,51,64,60,61,48,55);(1,4)(2,10)(3,15)(5,22)(6,25)(7,24)(8,23)(9,29)(11,35)(12,37)(13,20)(14,36)(16,42)(17,44)(18,21)(19,43)(26,54)(27,53)(28,52)(30,55)(31,34)(32,41)(33,49)(38,48)(39,60)(40,51)(45,50)(46,63)(47,62)(56,61)(57,64)(58,59);(1,5)(2,11)(3,16)(4,22)(6,27)(7,26)(8,28)(9,30)(10,35)(12,40)(13,38)(14,39)(15,42)(17,46)(18,45)(19,47)(20,48)(21,50)(23,52)(24,54)(25,53)(29,55)(31,58)(32,56)(33,57)(34,59)(36,60)(37,51)(41,61)(43,62)(44,63)(49,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271703);