Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| LCS | G ▷ C23 ▷ 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,12)(8,18,19,11)(9,13)(10,14)(15,26)(16,24,28,22)(17,25)(20,31,32,23)(21,27)(29,35)(30,34,36,33);(1,2,3,4)(6,11)(7,15)(12,24)(13,23)(14,18)(16,28)(17,29)(22,26)(25,34)(27,31)(30,36)(33,35);(5,7,9,17,10,15,21,29)(6,12,13,25,14,26,27,35)(8,16,20,30,19,28,32,36)(11,22,23,33,18,24,31,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,g^2=i,b^a=bd,d^a=dh,h^a=hi,c^b=ci,e^b=eh,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240802, #72257225, #72311919, … |
| maximal normal subgroup of | #72240802, #72257225, #72311919, … |
| maximal quotient of | #72240806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53549);