Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3140 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,21,24)(19,20,23,22)(25,26,29,32)(27,31,28,30);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,31)(29,32);(3,7)(5,8)(10,13)(14,15)(19,23)(20,22)(25,27,29,28)(26,30,32,31);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,23)(25,28,29,27)(26,31,32,30);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,24)(21,22)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232633, #72232686, #72232701, … |
| maximal normal subgroup of | #72232633, #72232686, #72232701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3492843);