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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,12,13)(11,14,16,15)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,31,32,28);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,30)(31,32);(3,7)(6,8)(10,13)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,29,32)(26,28,30,31);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,28,29,31)(26,27,30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,21)(20,24)(22,23)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fhi,d^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 | #72266510, #72266513, #72266516, … |
| maximal normal subgroup of | #72266510, #72266513, #72266516, … |
| maximal quotient of | #72334844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3443415);