Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
3630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,14)(12,15)(13,16)(18,22,24,21)(20,23);(5,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23);(2,4)(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240192, #72240193, #72240398, … |
| maximal normal subgroup of | #72240192, #72240193, #72240398, … |
| maximal quotient of | #72250057, #72250064, #72266029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30777);