Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3086 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)(9,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,20,24)(25,26)(27,30)(28,29)(31,32);(4,8)(6,7)(10,12)(11,16)(17,19,22,20)(18,21,23,24)(25,27,29,32)(26,30,28,31);(2,3)(6,7)(10,12)(14,15)(18,23)(19,20)(26,28)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,21,23,24)(25,28,29,26)(27,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21)(18,19)(20,23)(22,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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 | #72219213, #72219214, #72219275 |
| maximal normal subgroup of | #72219213, #72219214, #72219275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143680);