Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3118 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)(3,6)(4,8)(5,7)(9,10,13,14)(11,12,15,16)(17,18,21,23)(19,20,24,22)(25,26)(27,32)(28,30)(29,31);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,15,16)(17,19,21,24)(18,22,23,20)(25,27,29,30)(26,28,31,32);(2,7)(3,8)(10,14)(11,15)(17,19,21,24)(18,20,23,22)(25,28,29,32)(26,30,31,27);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^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 | #72327832, #72327899, #72327928 |
| maximal normal subgroup of | #72327832, #72327899, #72327928 |
| maximal quotient of | #72329097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143892);