Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25770 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,28)(22,25)(23,26)(24,27);(6,9)(10,12)(13,15,16,23)(14,18,19,26)(17,21,24,22)(20,28,27,25);(1,2,3,4)(14,19)(15,21)(17,24)(18,25)(22,23)(26,28);(15,22)(17,24)(18,25)(20,27)(21,23)(26,28);(5,7,8,11)(6,10,9,12)(15,23)(18,26)(21,22)(25,28)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72256912, #72256919, … |
| maximal normal subgroup of | #72239609, #72256912, #72256919, … |
| maximal quotient of | #72239733, #72264383, #72264442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352065);