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 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,38,40)(35,41,42,36)(37,45,46,39)(43,44,48,47);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,19)(20,27)(21,29)(30,31)(35,42)(36,41)(43,48)(44,47);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,47)(46,48);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,39,40,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,c^2=i,d^2=gh,e^2=fg,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, 1871708);