Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,31)(29,32);(2,5)(6,8)(11,12)(14,16);(9,11,13,12)(10,14,15,16)(17,19,20,29,22,28,21,27)(18,23,30,31,25,32,24,26);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,32,26);(1,3,4,7)(2,6,5,8)(19,28)(23,32)(26,31)(27,29)Polycyclic presentation:
2^9|c^2=gi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72251216, #72251237, #72251242, … |
| maximal normal subgroup of | #72251216, #72251237, #72251242, … |
| maximal quotient of | #72308408, #72308429, #72308589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370687);