Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,19)(20,30,32,40)(21,25,34,28)(22,29,37,26)(24,36,27,33)(31,43,44,48)(35,42,46,39)(38,47,41,45);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,23,18)(20,27,32,24)(21,28,34,25)(22,26,37,29)(30,36,40,33)(31,41,44,38)(35,39,46,42)(43,47,48,45);(3,9)(6,13)(10,11)(14,15)(17,20)(18,24)(19,27)(21,31)(22,36)(23,32)(25,38)(26,40)(28,41)(29,30)(33,37)(34,44)(35,47)(39,48)(42,43)(45,46);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,34)(18,25,19,28)(20,31,32,44)(22,35,37,46)(24,38,27,41)(26,39,29,42)(30,43,40,48)(33,45,36,47);(2,8)(6,13)(7,16)(14,15)(21,34)(25,28)(31,44)(35,46)(38,41)(39,42)(43,48)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388686);