Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(18,22)(20,27)(23,28)(26,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,47)(43,46)(45,48);(9,10,13,16)(11,14,12,15)(18,22)(19,25)(24,29)(26,31)(34,39)(37,40)(41,44)(46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,42)(34,40,39,37)(36,45,47,43)(41,46,44,48);(1,2)(3,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,47)(34,41,39,44)(35,43,42,45)(37,46,40,48);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,40)(34,42,39,35)(36,46,47,48)(41,45,44,43)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=gi,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1769813);