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:
(1,2,8,15)(3,17,23,44)(4,25,28,46)(5,30,36,13)(6,11,37,14)(7,20,39,50)(9,16,18,40)(10,31,27,21)(12,33,43,26)(19,55,22,60)(24,52,49,51)(29,56,53,58)(32,61,54,48)(34,47,64,42)(35,57,63,45)(38,59,41,62);(1,3,18,49)(2,10,40,38)(4,20,53,14)(5,31,54,59)(6,12,39,22)(7,19,37,43)(8,23,9,24)(11,28,50,29)(13,44,61,52)(15,27,16,41)(17,48,51,30)(21,32,62,36)(25,35,58,64)(26,57,55,42)(33,45,60,47)(34,46,63,56);(2,11)(3,19)(5,32)(6,37)(7,39)(10,27)(12,24)(13,45)(14,15)(16,20)(17,51)(21,25)(22,23)(26,60)(30,57)(31,46)(33,55)(34,63)(35,64)(36,54)(38,41)(40,50)(42,48)(43,49)(44,52)(47,61)(56,59)(58,62);(1,4)(2,12)(3,20)(5,33)(6,38)(7,27)(8,28)(9,29)(10,39)(11,24)(13,46)(14,49)(15,43)(16,19)(17,35)(18,53)(21,57)(22,40)(23,50)(25,30)(26,36)(31,45)(32,55)(34,52)(37,41)(42,62)(44,63)(47,59)(48,58)(51,64)(54,60)(56,61)Polycyclic presentation:
2^9|a^2=h,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, 151736);