Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,17)(8,14)(9,16)(10,12)(11,15)(13,19)(18,32)(20,29)(21,30)(22,26)(23,25)(24,28)(27,35)(31,34)(33,36);(1,2,3,4)(6,12)(13,25)(14,26)(15,16)(17,29)(27,36)(28,30)(31,32);(5,7)(6,13)(8,18)(9,19)(10,20)(11,23)(12,25)(14,27)(15,17)(16,29)(21,33)(22,34)(24,35)(26,36)(28,31)(30,32);(5,8)(6,14)(7,18)(9,21)(10,22)(11,24)(12,26)(13,27)(15,28)(16,30)(17,31)(19,33)(20,34)(23,35)(25,36)(29,32);(5,9,10,11)(6,15,12,16)(7,19,20,23)(8,21,22,24)(13,17,25,29)(14,28,26,30)(18,33,34,35)(27,31,36,32)Polycyclic presentation:
2^9|b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,g^c=gi,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311254, #72311281, #72312064, … |
| maximal normal subgroup of | #72311254, #72311281, #72312064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536168);