Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,20)(21,24)(22,23);(2,4)(9,11)(10,15,18,23)(12,20)(13,22)(14,21)(16,19,17,24);(9,10)(11,19,21,24)(12,16)(13,17)(14,18)(15,20,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239267, #72239270, #72265536, … |
| maximal normal subgroup of | #72239267, #72239270, #72265536, … |
| maximal quotient of | #72240290, #72260929, #72279486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31104);