Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #537 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,23,28,24,30,22,29)(19,31,34,25,35,42,41,43)(20,36,37,46,33,45,38,26)(21,27,40,47,32,44,39,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,34,24,35,22,41)(18,25,29,31,30,43,28,42)(20,21,38,39,33,32,37,40)(26,44,36,48,46,27,45,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,24,33)(18,26,30,46)(19,32,35,21)(22,37,23,38)(25,27,43,44)(28,45,29,36)(31,47,42,48)(34,40,41,39);(3,10)(6,9)(11,16)(14,15)(17,21)(18,27)(19,33)(20,35)(22,39)(23,40)(24,32)(25,26)(28,47)(29,48)(30,44)(31,45)(34,38)(36,42)(37,41)(43,46);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,30,29)(19,34,35,41)(20,37,33,38)(21,39,32,40)(25,31,43,42)(26,45,46,36)(27,47,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089615);