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 | #60926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 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)(17,18,21,38,23,27,22,28)(19,29,33,24,32,46,34,42)(20,26,39,25,37,44,36,43)(30,40,31,45,35,41,47,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,22,34,23,32,21,33)(18,24,38,46,27,42,28,29)(20,35,39,47,37,30,36,31)(25,41,26,45,43,40,44,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,37)(18,25,27,43)(19,30,32,35)(21,36,22,39)(24,40,42,41)(26,28,44,38)(29,45,46,48)(31,33,47,34);(3,10)(6,9)(11,16)(14,15)(17,20)(18,26)(19,31)(21,39)(22,36)(23,37)(24,41)(25,38)(27,44)(28,43)(29,45)(30,34)(32,47)(33,35)(40,42)(46,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,27)(19,32)(25,43)(26,44)(28,38)(30,35)(31,47)(33,34)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,e^2=hi,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, 7089620);