Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,23,18,21,16,22)(15,24,25,20,27,28,26,19);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,21)(22,24)(23,28);(2,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,28)(23,24);(14,21)(19,20)(22,23)(24,28);(13,16,18,17)(14,22,21,23)(15,25,27,26)(19,24,20,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224731, #72224734, #72237210, … |
| maximal normal subgroup of | #72224731, #72224734, #72237210, … |
| maximal quotient of | #72224711, #72238947, #72239617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339607);