Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388466 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14520 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1173 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)(4,5,19,22)(6,23)(7,11)(8,15)(9,21,18,24)(10,25)(13,14,17,30)(16,29)(20,27,31,28)(26,32);(1,3,6,15,7,16,25,32)(2,8,10,12,11,26,23,29)(4,17,18,28,19,13,9,27)(5,20,21,30,22,31,24,14);(1,4,6,9,7,19,25,18)(2,5,10,24,11,22,23,21)(3,13,15,28,16,17,32,27)(8,20,12,14,26,31,29,30);(1,5,7,22)(2,9,11,18)(3,14,16,30)(4,10,19,23)(6,24,25,21)(8,13,26,17)(12,28,29,27)(15,31,32,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339020, #72339032, #72339037 |
| maximal normal subgroup of | #72339020, #72339032, #72339037 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295663);