Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
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 | |
6959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,15,16,14)(17,18)(19,21)(20,22)(23,24);(2,6)(7,8)(10,13)(14,15)(19,23)(21,24);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,21)(20,23)(22,24);(11,16)(14,15)(18,22)(19,23);(10,13)(14,15)(19,23)(21,24);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=gh,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 | #72258185, #72278298, #72278655, … |
| maximal normal subgroup of | #72258185, #72278298, #72278655, … |
| maximal quotient of | #72335296, #72335366, #72336381 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609467);