Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,15)(12,18)(13,17)(14,19)(16,20)(21,24)(22,23);(1,2,4,3)(5,6)(7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,17)(18,23)(19,24);(1,3,4,2)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,17)(18,24)(19,23);(6,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,17)(18,23)(19,24);(9,12,14,13)(10,17,19,18)(11,20,21,22)(15,23,24,16)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315629, #72315645, #72315800, … |
| maximal normal subgroup of | #72315629, #72315645, #72315800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346036);