Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,16,12,14)(17,18,21,20)(19,23,24,22)(25,26,29,31)(27,32,30,28);(1,2)(3,5)(4,7)(6,8)(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,28)(29,30)(31,32);(2,3)(6,7)(11,12)(14,16)(18,20)(22,23)(27,30)(28,32);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,14,15,16)(17,20,21,18)(19,22,24,23)(25,28,29,32)(26,30,31,27);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72334189, #72334197, #72334234 |
| maximal normal subgroup of | #72334189, #72334197, #72334234 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4381989);