Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
2346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10)(11,17,19,14)(12,15,20,22)(13,16)(18,23)(21,24);(1,2)(3,4)(5,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,4)(9,12)(10,15,16,22)(11,18)(13,20)(14,21,17,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265632, #72265637, #72265641, … |
| maximal normal subgroup of | #72265632, #72265637, #72265641, … |
| maximal quotient of | #72266076, #72266077 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30921);