Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
5739 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)(3,6)(4,5)(7,8)(10,13)(11,15)(19,23)(22,24);(2,5)(6,8)(18,21)(22,24);(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23);(10,13)(14,16)(17,19,20,23)(18,22,21,24);(9,11,12,15)(10,14,13,16)(18,21)(22,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304025 |
| maximal normal subgroup of | #72304025 |
| maximal quotient of | #72304081, #72304094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609813);