Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3040 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,28)(21,26)(22,27);(2,4)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,18,17)(14,21,22,20)(15,24,23,25)(19,26,27,28);(2,4)(15,23)(19,27)(24,25)(26,28)Polycyclic presentation:
2^9|b^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72256825, #72256855, #72256989, … |
| maximal normal subgroup of | #72256825, #72256855, #72256989, … |
| maximal quotient of | #72307823, #72311345, #72311519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484080);