Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14181 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,12,6,14)(3,9,17,24,16,25,5,11)(4,18,20,10,21,28,22,27)(15,29,31,32,30,26,19,23);(1,3,6,17,8,16,7,5)(2,9,13,11,12,25,14,24)(4,19,20,30,21,31,22,15)(10,26,18,23,27,29,28,32);(1,4)(2,10)(3,15)(5,19)(6,22)(7,20)(8,21)(9,23)(11,26)(12,27)(13,28)(14,18)(16,30)(17,31)(24,29)(25,32);(1,5,7,16,8,17,6,3)(2,11,14,9,12,24,13,25)(4,19,20,30,21,31,22,15)(10,26,18,23,27,29,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319366, #72319471, … |
| maximal normal subgroup of | #72319143, #72319366, #72319471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283684);