Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(13,14,17,22,16,21,25,28,18,23,26,15,20,19,27,24);(2,4)(5,6,7,9,8,10,11,12)(14,19,23,21)(15,24,22,28)(17,26)(25,27);(13,15,25,14,20,28,17,19,18,22,27,23,16,24,26,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=cg,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225535, #72225537, D4×#1313, … |
| maximal normal subgroup of | #72225535, #72225537, D4×#1313, … |
| maximal quotient of | #72225532, #72225535, C2×#129049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36245);