Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427666 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2527 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15371 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
868 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)(3,11,14,7)(4,15)(5,18)(6,9)(8,17)(10,21)(12,22,19,25)(13,23,30,27)(16,24)(20,26)(28,32,29,31);(3,12,14,19)(7,22,11,25)(13,29,30,28)(23,32,27,31);(1,3)(2,7)(4,13)(5,19)(6,14)(8,23)(9,11)(10,25)(12,21)(15,27)(16,28)(17,30)(18,22)(20,29)(24,31)(26,32);(1,4)(2,8,9,15)(3,13)(5,20)(6,17)(7,23,11,27)(10,26,18,24)(12,28)(14,30)(16,21)(19,29)(22,31,25,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,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 | #72286050, #72286106, #72286246, … |
| maximal normal subgroup of | #72286050, #72286106, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334862);