Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2528 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)(9,10,12,14)(11,15,16,13)(17,18)(20,21)(22,24);(9,11)(10,13)(12,16)(14,15)(18,21)(19,22)(23,24);(2,4)(17,19)(18,22)(20,23)(21,24);(6,8)(11,16)(13,15)(17,18,20,21)(19,22,23,24)Polycyclic presentation:
2^9|a^2=eh,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72305537, #72305576, #72305635, … |
| maximal normal subgroup of | #72305537, #72305576, #72305635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356742);