Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,12)(11,13,16,21)(14,17,24,22)(18,19,23,20);(1,2,3,4)(9,11,15,16)(10,13,12,21)(14,19,24,20)(17,23,22,18);(6,8)(9,12,15,10)(11,18,16,23)(13,19,21,20)(14,17,24,22);(9,13,15,21)(10,16,12,11)(14,23,24,18)(17,20,22,19);(11,19)(13,23)(14,24)(16,20)(17,22)(18,21)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285009, #72307186, #72307195, … |
| maximal normal subgroup of | #72285009, #72307186, #72307195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481339);