Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4790 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)(10,16)(17,23)(18,24)(19,20);(2,4)(9,10)(11,18)(12,23)(13,20)(14,16)(15,19)(17,21)(22,24);(6,8)(9,11,14,22)(10,17,16,23)(12,13,21,15)(18,20,24,19);(9,12)(10,18)(11,13)(14,21)(15,22)(16,24)(17,20)(19,23);(9,13,14,15)(10,19,16,20)(11,21,22,12)(17,18,23,24)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313380, #72313459, #72313514, … |
| maximal normal subgroup of | #72313380, #72313459, #72313514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344885);