Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470037 |
| 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 | #60808 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2840 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)(18,23)(21,32)(24,37)(25,36)(26,33)(28,34)(31,43)(35,39)(38,47)(40,44)(42,45)(46,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18)(19,24)(20,25)(21,33)(22,26)(23,32)(27,38)(28,35)(29,39)(30,40)(31,44)(34,37)(36,43)(41,48)(42,46)(45,47);(3,9)(6,13)(10,11)(14,15)(17,19,22,29)(18,24,26,39)(20,27,30,41)(21,34,32,28)(23,35,33,37)(25,38,40,48)(31,45,43,42)(36,46,44,47);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20,22,30)(18,25,26,40)(19,27,29,41)(21,31,32,43)(23,36,33,44)(24,38,39,48)(28,42,34,45)(35,46,37,47);(2,8)(6,13)(7,16)(14,15)(20,30)(25,40)(27,41)(31,43)(36,44)(38,48)(42,45)(46,47)Polycyclic presentation:
2^9|c^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388822);