Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351559 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(10,15)(11,19)(12,13)(17,18)(20,21)(22,24);(3,7)(5,8)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,23)(21,22);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(11,20)(16,23)(19,21)(22,24);(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,24,23,22)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257726, #72340534, #72340538, … |
| maximal normal subgroup of | #72257726, #72340534, #72340538, … |
| maximal quotient of | #72257867, #72279302, #72279322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258759);