Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14016 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,3,6,11,7,12)(4,15,17,25,18,13,23,9)(5,19,21,26,20,10,22,14)(16,28,31,32,30,27,29,24);(1,3,8,11,6,12,7,2)(4,13,17,9,18,15,23,25)(5,14,21,19,20,26,22,10)(16,27,31,24,30,28,29,32);(1,4,6,18)(2,9)(3,13)(5,16,20,30)(7,17,8,23)(10,24)(11,25)(12,15)(14,27)(19,32)(21,29,22,31)(26,28);(1,5)(2,10)(3,14)(4,16)(6,20)(7,22)(8,21)(9,24)(11,19)(12,26)(13,27)(15,28)(17,31)(18,30)(23,29)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319325, #72319375, #72319471, … |
| maximal normal subgroup of | #72319325, #72319375, #72319471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275424);