Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13970 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,33,18,41,17,9)(4,20,23,36,24,10,30,37)(5,25,27,38,28,56,26,11)(15,42,44,34,21,39,49,31)(16,45,48,61,47,32,46,40)(22,52,53,50,54,55,57,35)(43,58,63,62,51,59,64,60);(1,3)(2,9,13,33)(4,21,24,15)(5,16)(6,17)(7,19)(8,18)(10,34)(11,32,38,45)(12,14,29,41)(20,31)(22,51,54,43)(23,49,30,44)(25,40,56,61)(26,46)(27,48)(28,47)(35,59)(36,42)(37,39)(50,58)(52,60)(53,64,57,63)(55,62);(1,4)(2,10)(3,15)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,20)(14,39)(16,43)(17,44)(18,21)(19,49)(25,55)(26,53)(27,57)(28,54)(29,37)(32,58)(33,34)(38,50)(40,60)(41,42)(45,59)(46,63)(47,51)(48,64)(52,56)(61,62);(1,5,6,26,8,28,7,27)(2,11,12,25,13,38,29,56)(3,16,17,46,18,47,19,48)(4,22,23,53,24,54,30,57)(9,32,14,40,33,45,41,61)(10,35,36,55,20,50,37,52)(15,43,44,63,21,51,49,64)(31,58,39,60,34,59,42,62)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272250);