Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| 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 | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2325 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)(3,9,17,31,18,32,27,39)(4,19,24,10,22,36,23,35)(5,11,25,37,26,38,28,40)(15,41,43,33,20,49,47,29)(16,30,46,58,45,57,44,56)(21,51,53,60,52,48,54,34)(42,62,64,55,50,61,63,59);(1,3,7,17,8,18,6,27)(2,9,13,31,14,32,12,39)(4,20,24,47,22,15,23,43)(5,16,25,46,26,45,28,44)(10,33,36,49,35,29,19,41)(11,30,37,58,38,57,40,56)(21,50,53,63,52,42,54,64)(34,59,51,62,60,55,48,61);(1,4)(2,10)(3,15)(5,21)(6,24)(7,23)(8,22)(9,29)(11,34)(12,36)(13,19)(14,35)(16,42)(17,47)(18,20)(25,54)(26,52)(27,43)(28,53)(30,55)(31,49)(32,33)(37,48)(38,60)(39,41)(40,51)(44,64)(45,50)(46,63)(56,61)(57,59)(58,62);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,26)(9,30)(10,34)(12,40)(13,37)(14,38)(15,42)(17,46)(18,45)(19,48)(20,50)(22,52)(23,54)(24,53)(27,44)(29,55)(31,58)(32,57)(33,59)(35,60)(36,51)(39,56)(41,61)(43,64)(47,63)(49,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 271724);