Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14768 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
916 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)(4,16)(5,19)(6,9)(8,18)(10,20)(12,21,15,23)(13,22,27,25)(17,24)(26,32,28,31)(29,30);(1,3)(2,7)(4,13)(5,12)(6,14)(8,22)(9,11)(10,21)(15,20)(16,25)(17,26)(18,27)(19,23)(24,31)(28,30)(29,32);(3,12,14,15)(7,21,11,23)(13,26,27,28)(22,31,25,32);(1,4)(2,8,9,16)(3,13)(5,17)(6,18)(7,22,11,25)(10,24,19,29)(12,26)(14,27)(15,28)(20,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350, #72286106, #72286218, … |
| maximal normal subgroup of | #72280350, #72286106, #72286218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305969);