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 |
| prime radical | #32911 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3114 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)(5,6,10,9)(7,11,8,12)(13,14,17,20)(15,19,22,26)(16,18,24,21)(23,25,28,27);(5,7,10,8)(6,11,9,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(5,6,10,9)(7,12,8,11)(13,15,17,22)(14,19,20,26)(16,23,24,28)(18,25,21,27);(5,8,10,7)(6,12,9,11)(15,22)(19,26)(23,28)(25,27);(5,9,10,6)(7,11,8,12)(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257549, #72257597, #72263870, … |
| maximal normal subgroup of | #72257549, #72257597, #72263870, … |
| maximal quotient of | #72305504, #72305594, #72305683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534469);