Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(11,15)(13,16)(18,21)(19,23);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,22,21,24);(2,5)(6,8);(1,2)(3,6)(4,5)(7,8);(1,3)(2,6)(4,7)(5,8)Polycyclic presentation:
2^9|b^2=gh,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| Fitting subgroup of | #72258842 |
| maximal subgroup of | #72258842, #72258908 |
| maximal normal subgroup of | #72258842, #72258908 |
| maximal quotient of | #72258789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538150);