Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #24364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12);(6,9)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,25)(27,28);(2,4)(13,15,18,23)(14,19,21,28)(16,25,26,22)(17,27,20,24);(5,7,8,11)(6,10,9,12)(13,16,18,26)(14,20,21,17)(15,19,23,28)(22,24,25,27);(14,16)(15,23)(17,20)(19,22)(21,26)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256959, #72257018, #72257035, … |
| maximal normal subgroup of | #72256959, #72257018, #72257035, … |
| maximal quotient of | #72264158, #72264211 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483404);