Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220244 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(17,18)(19,23)(20,28,30,24)(21,25)(22,26)(27,32,29,31);(1,2,5,9)(3,7,11,15)(4,8,12,6)(10,14,16,13)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(2,8)(4,12)(6,9)(7,14)(10,16)(13,15)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,g^c=gi,e^d=eh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72268356, #72268359, #72268365, … |
| maximal normal subgroup of | #72268356, #72268359, #72268365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127457);