Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2252 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)(7,11)(8,10)(9,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,27)(26,28);(2,3)(5,7,8,12)(6,9,10,11)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,26,21,27);(1,2,4,3)(7,12)(9,11)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(6,10)(9,11)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(1,3,4,2)(14,20)(15,23)(19,21)(24,28)Polycyclic presentation:
2^9|b^2=f,c^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285000, #72285002, #72285539 |
| maximal normal subgroup of | #72285000, #72285002, #72285539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717760);