Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6,7,8)(9,10,13,15)(11,16,12,14)(18,19)(20,24);(1,2)(3,4)(9,11,13,12)(10,14,15,16)(18,19)(22,23);(2,4)(10,15)(14,16)(17,18,21,19)(20,22,24,23);(5,6)(7,8)(9,12,13,11)(10,16,15,14)(17,19,21,18)(20,23,24,22);(1,2)(3,4)(9,10)(11,14)(12,16)(13,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fg,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72331499, #72331524, #72331547 |
| maximal normal subgroup of | #72331499, #72331524, #72331547 |
| maximal quotient of | #72332889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2025909);