Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(13,14)(15,18)(16,19)(17,22)(20,25)(21,24)(23,27)(26,28);(10,12)(14,15)(19,21)(20,22)(26,27);(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)Polycyclic presentation:
2^9|c^2=i,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72310672, #72310684, #72310689, … |
| maximal normal subgroup of | #72310672, #72310684, #72310689, … |
| maximal quotient of | #72314469, #72315526, #72315665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424610);