Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,18)(3,12,15,8,14,23,19,21)(4,9,16,26,17,25,20,24)(13,28,31,32,29,27,30,22);(8,21)(12,23)(22,32)(27,28);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,22)(10,23)(11,21)(12,18)(16,31)(17,29)(20,30)(24,27)(25,32)(26,28);(1,4,6,16,7,17,5,20)(2,9,10,26,11,25,18,24)(3,13,15,31,14,29,19,30)(8,22,23,28,21,32,12,27)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216071, #72216123, … |
| maximal normal subgroup of | #72216029, #72216071, #72216123, … |
| maximal quotient of | #72237188, C2×#375780, #72252194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282976);