Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18)(19,21)(20,23)(22,24);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,18)(20,23,24,22);(9,11)(10,13)(12,16)(14,15)(18,19)(22,23);(1,3,5,4)(2,7,8,6)(10,14)(13,15)(17,20,21,24)(18,22,19,23);(11,16)(13,15);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|b^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304031, #72316535, C2×#7639609 |
| maximal normal subgroup of | #72304031, #72316535, C2×#7639609 |
| maximal quotient of | #72304114, #72316543, C2×#7639609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546805);