Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,15)(12,17)(13,18,22,24)(14,19)(16,20,23,21);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,19,17)(13,20,22,21)(16,18,23,24);(2,5)(6,8)(9,11,14,12)(10,16,19,23)(13,21,22,20)(15,18,17,24);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,19,15)(13,21,22,20)(16,24,23,18)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fi,g^c=ghi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72312188, #72312201, #72312208, … |
| maximal normal subgroup of | #72312188, #72312201, #72312208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403729);