Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #381 |
| 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 | |
3570 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)(11,14,18,22)(12,19,20,15)(13,16)(17,23)(21,24);(5,6)(7,8)(9,11)(10,14,16,22)(12,17)(13,18)(15,21,19,23)(20,24);(2,4)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,d^b=di,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240192, #72240400, #72240404, … |
| maximal normal subgroup of | #72240192, #72240400, #72240404, … |
| maximal quotient of | #72266035, #72309368, #72309370 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30723);