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 | |
2912 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)(9,10,12,16)(11,15,19,14)(13,17,21,24)(18,23,20,22);(1,2,3,4)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,21)(23,24);(6,8)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20);(9,11,12,19)(10,15,16,14)(13,18,21,20)(17,23,24,22);(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,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 | #72307195, #72307208, #72307324 |
| maximal normal subgroup of | #72307195, #72307208, #72307324 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481337);