Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
3685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(26,29)(30,32);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,23,20)(27,31)(30,32);(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,6,8,7)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27)(26,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72328577, #72328638, #72328661 |
| maximal normal subgroup of | #72328577, #72328638, #72328661 |
| maximal quotient of | #72330220 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2528338);