Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,23,27)(3,8,14,5,10,21,26,20)(4,15,19,28,18,9,22,25)(13,29,30,17,16,24,31,32);(1,3,6,14,7,10,23,26)(2,8,12,5,11,21,27,20)(4,16,19,31,18,13,22,30)(9,24,25,32,15,29,28,17);(1,4)(2,9)(3,13)(5,17)(6,22)(7,18)(8,24)(10,16)(11,15)(12,28)(14,31)(19,23)(20,32)(21,29)(25,27)(26,30);(1,5)(2,10)(3,11)(4,17)(6,21)(7,20)(8,23)(9,16)(12,26)(13,15)(14,27)(18,32)(19,24)(22,29)(25,31)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240440, #72240443, #72240459, … |
| maximal normal subgroup of | #72240440, #72240443, #72240459, … |
| maximal quotient of | #72251261, C2×#364534, #72312175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271730);