Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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 | |
2836 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)(10,12)(15,22)(16,17)(18,19);(1,2,3,4)(9,10)(11,18)(12,13)(14,16)(15,21)(17,23)(19,20)(22,24);(6,8)(9,11,13,20)(10,15,12,22)(14,24,23,21)(16,18,17,19);(9,12)(10,13)(11,19)(14,17)(15,24)(16,23)(18,20)(21,22);(10,16)(12,17)(14,23)(15,18)(19,22)(21,24)Polycyclic presentation:
2^9|b^2=fi,c^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250525, #72256912, #72256948 |
| maximal normal subgroup of | #72250525, #72256912, #72256948 |
| maximal quotient of | #72265198 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481056);