Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | #61332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 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)(17,18,22,29,24,28,23,40)(19,30,33,42,32,21,27,25)(20,34,38,46,37,48,39,26)(31,43,45,36,35,44,47,41);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,32)(18,25,28,42)(20,35,37,31)(21,40,30,29)(22,27,23,33)(26,44,46,43)(34,36,48,41)(38,45,39,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,39,24,37,22,38)(18,26,29,34,28,46,40,48)(19,31,33,45,32,35,27,47)(21,41,42,44,30,36,25,43);(3,10)(6,9)(11,16)(14,15)(17,21,22,25,24,30,23,42)(18,27,29,19,28,33,40,32)(20,36,38,44,37,41,39,43)(26,45,34,35,46,47,48,31);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,28)(19,32)(20,37)(27,33)(29,40)(36,41)(38,39)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093925);