Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2796 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)(10,14)(12,20)(15,22)(16,21)(17,23)(19,24);(6,7)(9,10)(11,17)(12,21)(13,16)(14,20)(15,18)(19,22)(23,24);(1,2,3,4)(5,6,8,7)(9,11,13,18)(10,15,16,17)(12,19,20,24)(14,22,21,23);(5,7,8,6)(10,16)(14,21)(15,17)(22,23);(11,18)(15,17)(19,24)(22,23)Polycyclic presentation:
2^9|c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316405, #72316431, #72316442 |
| maximal normal subgroup of | #72316405, #72316431, #72316442 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372686);