Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(18,20)(19,23)(25,26,30,29)(27,28,32,31);(2,5)(3,8)(10,14)(12,15)(17,18)(19,24)(20,21)(22,23)(25,27,30,32)(26,31,29,28);(3,8)(6,7)(10,14)(12,15)(18,20)(22,24)(26,29)(27,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,28)(26,27)(29,32)(30,31);(1,3,4,8)(2,6,5,7)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,22,23,24)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300778, #72300782, #72300802, … |
| maximal normal subgroup of | #72300778, #72300782, #72300802, … |
| maximal quotient of | #72329930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3782876);