Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,18,14,16,12,17)(11,19,20,15,22,24,21,23);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(10,16)(15,23)(17,18)(19,24);(9,12,14,13)(10,17,16,18)(11,20,22,21)(15,24,23,19)Polycyclic presentation:
2^9|a^2=h,b^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237210, #72237226, #72237250, … |
| maximal normal subgroup of | #72237210, #72237226, #72237250, … |
| maximal quotient of | #72238944, #72238950, #72239440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278604);