Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1570 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)(19,27)(21,28)(24,37)(25,35)(29,42)(30,43)(31,45)(33,44)(34,39)(38,47)(40,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,21,26,22,23,28,36)(19,24,27,39,30,34,43,37)(20,25,33,41,32,35,44,46)(29,38,42,48,31,40,45,47);(3,9)(6,13)(10,11)(14,15)(17,19,28,43,22,30,21,27)(18,24,36,37,23,34,26,39)(20,31,44,42,32,29,33,45)(25,40,46,48,35,38,41,47);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,35)(24,38)(26,41)(27,42)(28,44)(30,31)(34,40)(36,46)(37,47)(39,48)(43,45);(2,8)(6,13)(7,16)(14,15)(19,30)(20,32)(24,34)(25,35)(27,43)(33,44)(37,39)(41,46)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391815);