Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| 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 | #63003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,16)(7,17,20,29)(8,13,22,28)(9,15,10,14)(12,23,26,19)(18,32,35,31)(21,30,24,27)(25,33,36,34);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,23)(11,20)(13,25)(14,17)(15,29)(16,26)(21,33)(22,35)(24,34)(27,31)(28,36)(30,32);(2,4)(5,8)(6,13)(7,18)(9,21)(10,24)(11,22)(12,25)(14,27)(15,30)(16,28)(17,31)(19,33)(20,35)(23,34)(26,36)(29,32);(1,2)(3,4)(5,9,11,10)(6,14,16,15)(7,19,20,23)(8,21,22,24)(12,17,26,29)(13,27,28,30)(18,33,35,34)(25,31,36,32);(2,4)(5,9,11,10)(6,14,16,15)(7,19,20,23)(8,21,22,24)(12,17,26,29)(13,27,28,30)(18,33,35,34)(25,31,36,32)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72311366, #72311454, #72312230, … |
| maximal normal subgroup of | #72311366, #72311454, #72312230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480893);