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 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,15)(17,18,21,19)(20,22,24,23)(27,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,14)(18,19)(22,23)(25,26,28,30)(27,29,31,32);(2,6)(5,7)(10,14)(12,16)(17,18,21,19)(20,23,24,22)(25,27)(26,29)(28,31)(30,32);(1,3)(2,7)(4,8)(5,6)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,22,24,23)(25,26)(27,29)(28,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fgh,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 | #72326796, #72326808, #72326923 |
| maximal normal subgroup of | #72326796, #72326808, #72326923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3460908);