Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4789 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)(9,10,14,18,15,17,22,24)(11,19,16,23,20,12,21,13);(5,6)(7,8)(9,11)(10,13)(12,18)(14,21)(15,20)(16,22)(17,23)(19,24);(2,4)(6,8)(9,12,15,19)(10,16,17,21)(11,18,20,24)(13,22,23,14);(6,8)(9,13)(10,11)(12,22)(14,19)(15,23)(16,18)(17,20)(21,24);(10,17)(12,19)(13,23)(18,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315788, #72315817, #72315856, … |
| maximal normal subgroup of | #72315788, #72315817, #72315856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344977);