Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
3435 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)(4,6,8,7)(10,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,31)(30,32);(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,24,21)(25,27,30,29)(26,31,32,28);(2,3)(6,7)(10,13)(14,15)(18,24)(19,20)(25,28,30,31)(26,27,32,29);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,20,22,19)(18,23,24,21)(25,28)(26,27)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,19,24,20)(25,29,30,27)(26,31,32,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72326673, #72326677, #72326765 |
| maximal normal subgroup of | #72326673, #72326677, #72326765 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5660886);