Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3211 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)(13,14,16,21,17,20,24,18)(15,19,23,27,22,26,28,25);(5,6,8,9)(7,10,11,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(6,9)(7,11)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(5,7)(6,10)(8,11)(9,12)(14,20)(15,22)(18,21)(23,28);(6,9)(7,11)(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257505, #72257557, #72263870, … |
| maximal normal subgroup of | #72257505, #72257557, #72263870, … |
| maximal quotient of | #72305514, #72305586, #72305678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503320);