Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2686 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15549 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,17)(3,8,18,20,14,22,12,25)(4,9,19,23,15,24,16,26)(13,21,27,31,29,32,28,30);(3,12,14,18)(8,20,22,25)(13,28,29,27)(21,31,32,30);(1,3)(2,8)(4,13)(5,14)(6,18)(7,12)(9,21)(10,25)(11,22)(15,29)(16,27)(17,20)(19,28)(23,30)(24,32)(26,31);(1,4,5,15)(2,9)(3,13,14,29)(6,19,7,16)(8,21)(10,26)(11,24)(12,27,18,28)(17,23)(20,30)(22,32)(25,31)Polycyclic presentation:
2^9|a^2=fg,b^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319947, #72319976, #72320069, … |
| maximal normal subgroup of | #72319947, #72319976, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343291);