Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2239 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,24)(17,19)(20,22)(21,25)(23,27)(26,28);(2,4)(6,9)(10,12)(13,15,20,17)(14,18,22,19)(16,21,27,28)(23,26,24,25);(5,7,8,11)(6,10,9,12)(13,16,18,25,20,27,19,26)(14,21,17,24,22,28,15,23);(1,2,3,4)(13,17,19,14,20,15,18,22)(16,23,26,28,27,24,25,21);(6,9)(10,12)(13,18,20,19)(14,15,22,17)(16,25,27,26)(21,23,28,24)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257053, #72257096, #72257217, … |
| maximal normal subgroup of | #72257053, #72257096, #72257217, … |
| maximal quotient of | #72264827 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520906);