Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2546 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)(9,10,13,16,14,18,12,17)(11,19,21,15,22,24,23,20);(9,11,12,23,14,22,13,21)(10,15,17,19,18,20,16,24);(2,4)(9,10)(11,20)(12,16)(13,17)(14,18)(15,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238738, #72238743, #72238746, … |
| maximal normal subgroup of | #72238738, #72238743, #72238746, … |
| maximal quotient of | #72238865, #72238870, #72238879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30820);