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 | #13984 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1137 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)(3,7,12,21)(4,14,17,8)(5,10)(6,9)(11,27)(13,22,29,32)(15,31)(16,26,18,24)(19,25)(20,28)(23,30);(1,3)(2,7,9,21)(4,15,17,30)(5,13)(6,12)(8,23)(10,22,25,32)(11,16,28,18)(14,31)(19,29)(20,26)(24,27);(1,4)(2,8)(3,11)(5,18)(6,17)(7,20)(9,14)(10,26)(12,28)(13,30)(15,29)(16,19)(21,27)(22,31)(23,32)(24,25);(1,3)(2,7,9,21)(4,11)(5,13)(6,12)(8,20,14,27)(10,22,25,32)(15,16)(17,28)(18,30)(19,29)(23,26,31,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,g^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,g^c=gi,h^c=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72286147, #72286176, … |
| maximal normal subgroup of | #72285150, #72286147, #72286176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273195);