Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15323 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1228 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)(3,13,16,8)(4,18)(5,12)(6,11)(7,10)(9,19)(14,22,17,24)(15,29,30,32)(20,27)(21,26)(23,31,25,28);(1,3)(2,8)(4,15)(5,14)(6,17)(7,16)(9,25)(10,13)(11,22)(12,24)(18,32)(19,28)(20,30)(21,31)(23,26)(27,29);(3,14,16,17)(8,22,13,24)(15,28,30,31)(23,29,25,32);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,18)(12,27)(13,25)(14,28)(16,30)(17,31)(22,29)(24,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280350, #72287080, #72287267, … |
| maximal normal subgroup of | #72280350, #72287080, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331804);