Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 | |
3010 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);(1,2,3,4)(9,10)(11,18)(12,15)(13,17)(14,19)(16,22)(20,24)(21,23);(6,8)(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24);(9,10)(11,18)(12,15)(13,17)(14,19)(16,22)(20,24)(21,23);(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256912, #72256948, #72257133 |
| maximal normal subgroup of | #72256912, #72256948, #72257133 |
| maximal quotient of | #72265282 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481038);