Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,26)(25,28);(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,17,18,19)(21,23,24,28)(22,25,26,27);(14,18)(17,19)(23,28)(25,27);(6,7)(10,11)(15,20)(17,19)(22,26)(23,28);(5,8,9,12)(6,10,7,11)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72260045, #72303917, #72303922, … |
| maximal normal subgroup of | #72260045, #72303917, #72303922, … |
| maximal quotient of | #72259307 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428144);