Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
658 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)(7,16,18,11)(8,20)(9,24)(10,13)(12,17,29,34)(14,22)(15,26)(19,28,25,32)(21,30,35,27)(23,33)(31,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,19)(10,18)(12,20)(13,16)(14,27)(15,28)(21,22)(23,34)(24,32)(25,26)(29,33)(30,31)(35,36);(2,4)(5,7)(6,12,13,29)(8,21)(9,25)(10,18)(11,20,16,33)(14,32,31,28)(15,27,24,30)(17,22)(19,26)(23,35)(34,36)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250449, #72250482, #72250485, … |
| maximal normal subgroup of | #72250449, #72250482, #72250485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55799);