Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348447 |
| 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 | |
3302 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)(12,13)(16,22)(17,21)(19,20)(23,24);(3,7)(5,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(11,18)(16,23)(19,20)(22,24);(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257658, #72279417, #72283124, … |
| maximal normal subgroup of | #72257658, #72279417, #72283124, … |
| maximal quotient of | #72264304, #72279297, #72279322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255648);