Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2819 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(10,11)(12,16)(18,22)(19,25)(20,27)(31,32);(6,8)(10,11)(14,15)(17,18,21,22)(19,23,25,30)(20,24,27,28)(26,31,29,32);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,28)(21,22)(24,27)(25,30)(26,32)(29,31);(1,2)(3,4)(9,11,13,10)(12,15,16,14)(17,19,21,25)(18,23,22,30)(20,29,27,26)(24,32,28,31);(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,26)(21,27)(22,28)(23,31)(25,29)(30,32)Polycyclic presentation:
2^9|b^2=f,c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3240831);