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 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,7,8)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,20,27,24)(21,29,30,25)(26,31,32,28)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(3,7)(6,8)(18,23)(20,24)(21,30)(26,32)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3,5,7)(2,6,4,8)(9,11,13,10)(12,15,16,14)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,2,5,4)(3,6,7,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,5,2)(3,8,7,6)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fghi,d^2=gh,e^2=fh,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, 1824350);