Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7758 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)(9,10,11,12)(14,18)(16,24)(19,21)(22,28);(2,4)(6,8)(13,14,17,18)(15,19,23,21)(16,20,24,25)(22,26,28,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=i,b^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 | #72304732, #72304741, #72304750, … |
| maximal normal subgroup of | #72304732, #72304741, #72304750, … |
| maximal quotient of | #72305095, #72305096, #72305108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423580);