Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | #3494 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1555 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,24)(20,33)(21,26)(22,28)(23,29)(25,32)(27,34)(30,39)(31,40)(35,48)(36,42)(37,43)(38,45)(41,47)(44,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,24,29,25)(20,34,31,22)(21,36,37,35)(26,42,43,48)(27,40,28,33)(30,46,47,38)(39,44,41,45);(17,20,22,19,23,31,34,32)(18,25,28,33,29,24,27,40)(21,35,38,30,37,36,46,47)(26,41,45,48,43,39,44,42);(2,7)(6,9)(8,15)(14,16)(21,37)(26,43)(30,47)(35,36)(38,46)(39,41)(42,48)(44,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,38,23,37,34,46)(18,26,28,45,29,43,27,44)(19,30,31,36,32,47,20,35)(24,39,40,42,25,41,33,48)Polycyclic presentation:
2^9|b^2=fi,c^2=h,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716876);