Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16207 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
670 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)(4,16)(5,8)(6,11)(7,14)(9,19)(10,21)(13,23)(15,24)(17,26)(18,27)(20,22)(25,30)(28,31)(29,32);(1,3)(2,7,11,12)(4,17,19,13)(5,20)(6,14)(8,24,10,22)(9,26)(15,21)(16,23)(18,28,30,29)(25,31)(27,32);(2,8,11,10)(7,22,12,24)(9,25,16,27)(23,31,26,32);(1,4,5,18,6,19,21,30)(2,9,10,27,11,16,8,25)(3,13,20,29,14,17,15,28)(7,23,22,31,12,26,24,32)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339020, #72339027, #72339028 |
| maximal normal subgroup of | #72339020, #72339027, #72339028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361093);