Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806815 |
| 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 | #525 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1534 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)(19,25)(20,23)(21,26)(22,27)(24,28)(29,33)(30,40)(31,41)(32,34)(35,38)(36,42)(37,43)(39,44)(45,47)(46,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,24,33)(18,25,28,29)(20,34,26,40)(21,30,23,32)(22,37,39,46)(27,43,44,48)(31,36,47,38)(35,41,42,45);(17,20,23,28,24,26,21,18)(19,29,30,34,33,25,32,40)(22,35,38,44,39,42,36,27)(31,45,37,48,47,41,46,43);(2,7)(6,9)(8,15)(14,16)(17,21,24,23)(18,26,28,20)(19,30,33,32)(22,36,39,38)(25,40,29,34)(27,42,44,35)(31,37,47,46)(41,43,45,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,21,36,24,39,23,38)(18,27,26,42,28,44,20,35)(19,31,32,46,33,47,30,37)(25,41,34,48,29,45,40,43)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=i,e^2=fhi,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=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716867);