Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,12)(8,13,19,11)(9,15)(10,14)(16,25)(17,24,29,22)(18,26)(20,27,32,23)(21,28)(30,35)(31,34,36,33);(1,2,3,4)(6,11,14,13)(7,16)(12,24,25,22)(15,23,28,27)(17,29)(18,30)(26,34,35,33)(31,36);(5,7,9,18,10,16,21,30)(6,12,15,26,14,25,28,35)(8,17,20,31,19,29,32,36)(11,22,23,33,13,24,27,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,g^2=i,b^a=bd,d^a=dh,f^a=fi,h^a=hi,c^b=ci,e^b=ehi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280249, #72281189, #72312373, … |
| maximal normal subgroup of | #72280249, #72281189, #72312373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53555);