Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2759 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16669 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
636 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)(3,7)(4,8,15,24)(5,17)(6,10)(9,19)(11,26)(12,23,14,21)(13,22)(16,30,29,25)(18,20)(27,31,28,32);(3,11,13,18)(7,20,22,26)(12,27,14,28)(21,31,23,32);(1,3)(2,7)(4,14,15,12)(5,18)(6,13)(8,23,24,21)(9,26)(10,22)(11,19)(16,27,29,28)(17,20)(25,31,30,32);(1,4,5,16,6,15,19,29)(2,8,9,25,10,24,17,30)(3,12,18,28,13,14,11,27)(7,21,26,32,22,23,20,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319342, #72319383, #72319453 |
| maximal normal subgroup of | #72319342, #72319383, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377124);