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 | |
5579 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,22)(21,24);(10,14)(11,16)(17,19)(18,21)(20,24)(22,23);(10,14)(13,15)(18,22)(21,23);(19,24)(21,23);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^9|a^2=g,b^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 | #72256991, #72280725, #72316533, … |
| maximal normal subgroup of | #72256991, #72280725, #72316533, … |
| maximal quotient of | #72281578, #72281583, #72282128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546799);