Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,22,23,24)(25,26,30,31)(27,32,28,29);(12,16)(14,15)(18,21)(19,23)(25,27)(26,29)(28,30)(31,32);(10,11)(14,15)(18,21)(22,24)(27,28)(29,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,23)(21,24)(25,28,30,27)(26,29,31,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,29)(26,27)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=f,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72326558, #72326607, #72326624 |
| maximal normal subgroup of | #72326558, #72326607, #72326624 |
| maximal quotient of | #72328890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5267060);