Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2665 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16826 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
655 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)(3,12,15,7)(4,8)(5,10)(6,9)(11,13)(14,20,28,23)(16,22,27,21)(17,24)(18,25)(19,26)(29,31,30,32);(1,3,13,27,5,15,6,16)(2,7,11,22,10,12,9,21)(4,14,17,29,18,28,19,30)(8,23,24,32,25,20,26,31);(1,4)(2,8)(3,14,15,28)(5,18)(6,19)(7,20,12,23)(9,26)(10,25)(11,24)(13,17)(16,30,27,29)(21,32,22,31);(2,9,10,11)(7,21,12,22)(8,24,25,26)(20,31,23,32)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319935, #72319982, #72320069, … |
| maximal normal subgroup of | #72319935, #72319982, #72320069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379779);