Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,22)(23,24)(25,26,28,29)(27,31,32,30);(10,13)(14,15)(18,22)(20,23)(25,26)(27,31)(28,29)(30,32);(11,16)(14,15)(18,22)(19,24)(26,29)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,23,22,20)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,23)(18,24,22,19)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=fhi,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232578, #72232630, #72232691, … |
| maximal normal subgroup of | #72232578, #72232630, #72232691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1156182);