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 | #25097 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12);(2,3)(13,14)(15,21)(16,24)(17,20)(18,22)(19,26)(23,27)(25,28);(1,2,4,3)(6,9)(10,12)(13,15,17,23)(14,18,20,28)(16,25,26,22)(19,27,24,21);(1,3,4,2)(14,19)(16,26)(18,27)(20,24)(21,28)(22,25);(5,7,8,11)(6,10,9,12)(14,20)(18,28)(19,24)(21,27)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=f,e^2=g,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 | #72257058, #72257116, #72257176, … |
| maximal normal subgroup of | #72257058, #72257116, #72257176, … |
| maximal quotient of | #72265013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363581);