Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9)(3,17)(4,21)(5,26)(6,28)(11,38)(13,16)(14,47)(15,22)(18,51)(19,24)(20,23)(25,27)(29,60)(30,58)(31,37)(32,41)(33,43)(34,48)(35,49)(36,53)(39,57)(40,55)(42,62)(46,63)(50,52)(54,59)(56,61);(1,2,10,34)(3,12,36,64)(4,22,5,13)(6,29,37,49)(7,15,8,16)(9,26,48,21)(11,39,62,18)(14,28,61,31)(17,33,53,63)(19,44,20,45)(23,46,24,43)(25,35,27,60)(30,42,52,38)(32,50,41,58)(40,51,55,57)(47,54,56,59);(1,3)(2,11)(4,23)(5,24)(6,30)(7,19)(8,20)(9,32)(10,36)(12,29)(13,40)(14,44)(15,42)(16,38)(17,26)(18,31)(21,53)(22,55)(25,58)(27,50)(28,39)(33,47)(34,62)(35,46)(37,52)(41,48)(43,60)(45,61)(49,64)(51,54)(56,63)(57,59);(1,3)(2,12)(4,24)(5,23)(6,18)(7,19)(8,20)(9,33)(10,36)(11,29)(13,46)(14,42)(15,44)(16,45)(17,21)(22,43)(25,57)(26,53)(27,51)(28,52)(30,31)(32,47)(34,64)(35,40)(37,39)(38,61)(41,56)(48,63)(49,62)(50,54)(55,60)(58,59)Polycyclic presentation:
2^9|b^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151735);