Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3481 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,6,9,16,22,5,10,14,23,17,24,27,32)(3,7,12,19,13,20,26,30,11,18,15,21,25,29,28,31);(2,7)(4,14)(8,19)(9,20)(10,18)(12,15)(16,27)(21,23)(22,30)(24,29)(26,28)(31,32);(3,11)(12,15)(13,25)(26,28)Polycyclic presentation:
2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=egi,f^b=fg,d^c=dg,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231603, #72231607, … |
| maximal normal subgroup of | #72231596, #72231603, #72231607, … |
| maximal quotient of | #72239959, #72280076, C2×#129613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36809);