Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6447056 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2420 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,17)(5,22)(6,10)(7,11)(8,15)(9,20)(13,26)(14,25)(16,24)(18,28)(19,31)(21,27)(23,29)(30,32);(1,3,5,16,7,15,6,14)(2,8,10,24,11,12,22,25)(4,18,20,30,19,29,21,13)(9,26,17,23,27,32,31,28);(1,4,5,20,7,19,6,21)(2,9,10,17,11,27,22,31)(3,13,14,29,15,30,16,18)(8,23,25,26,12,28,24,32);(3,14,15,16)(8,24,12,25)(13,29,30,18)(23,32,28,26);(4,19)(9,27)(13,30)(17,31)(18,29)(20,21)(23,28)(26,32)Polycyclic presentation:
2^9|b^2=g,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319923, #72319976, #72320044, … |
| maximal normal subgroup of | #72319923, #72319976, #72320044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354252);