Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,17)(11,12,15,22)(13,16,23,24)(18,20,19,21);(2,4)(9,11,14,15)(10,12,17,22)(13,18,23,19)(16,20,24,21);(6,8)(11,18)(12,20)(13,23)(15,19)(16,24)(21,22);(9,12)(10,15)(11,17)(13,20)(14,22)(16,19)(18,24)(21,23);(11,18)(12,20)(13,23)(15,19)(16,24)(21,22)Polycyclic presentation:
2^9|a^2=i,b^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 | #72313514, #72313649, #72313685 |
| maximal normal subgroup of | #72313514, #72313649, #72313685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480668);