Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357891 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,18)(11,19,21,22)(12,16,15,24)(13,20,23,17);(9,11,13,16,14,21,23,24)(10,15,17,19,18,12,20,22);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,12)(10,16)(11,20)(13,22)(14,15)(17,21)(18,24)(19,23);(11,21)(12,15)(16,24)(19,22)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214074, #72338415, #72338422, … |
| maximal normal subgroup of | #72214074, #72338415, #72338422, … |
| maximal quotient of | #72224784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265088);