Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(10,12)(14,15)(17,18,21,23,22,24,20,19);(9,10,13,12)(11,15,16,14)(17,19,20,24,22,23,21,18);(2,4)(9,11,13,16)(10,14,12,15)(18,23)(19,24)(20,21);(6,8)(9,12,13,10)(11,14,16,15)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264681, #72264688, #72264697, … |
| maximal normal subgroup of | #72264681, #72264688, #72264697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372041);