Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
3050 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)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,29)(30,32);(11,15)(14,16)(18,20)(19,24)(25,26)(27,31)(28,29)(30,32);(10,13)(14,16)(18,20)(22,23)(26,29)(30,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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 | #72328187, #72328210, #72328227, … |
| maximal normal subgroup of | #72328187, #72328210, #72328227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5063883);