Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4550 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,28)(17,24)(18,23)(19,22)(20,27)(21,26);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,26)(14,20,22,28)(16,18,27,17)(21,24,25,23);(7,11)(9,12)(13,16,19,27)(14,21,22,25)(15,17,26,18)(20,23,28,24);(7,11)(9,12)(15,26)(16,27)(20,28)(21,25);(6,10)(9,12)(14,22)(20,28)(21,25)(23,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257758, #72282433, #72315541, … |
| maximal normal subgroup of | #72257758, #72282433, #72315541, … |
| maximal quotient of | #72264200, #72316127, #72316134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377309);