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 | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3372 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,23,24,20)(25,26,29,32)(27,31,28,30);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,19,21,24)(18,20,22,23)(25,26)(27,30)(28,31)(29,32);(2,4)(6,7)(10,12)(14,15)(19,24)(20,23)(25,27,29,28)(26,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28,29,27)(26,31,32,30);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,19,22,24)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=hi,d^2=hi,e^2=fhi,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 | #72236953, #72327598, #72327628, … |
| maximal normal subgroup of | #72236953, #72327598, #72327628, … |
| maximal quotient of | #72330314, #72330517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3764250);