Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,26)(9,16)(10,18)(11,17)(12,19)(13,20)(14,22)(21,32)(23,33)(24,34)(25,35)(27,30)(28,31)(29,36);(1,2)(3,4)(5,7)(6,14)(8,27)(9,21)(10,28)(11,22)(12,24)(13,25)(15,18)(16,31)(17,34)(19,33)(20,26)(23,29)(30,32)(35,36);(2,4)(5,8)(6,15)(7,10)(9,23)(11,24)(12,28)(13,22)(14,30)(16,32)(17,26)(18,33)(19,34)(20,35)(21,27)(25,29)(31,36);(5,9,13,12)(6,16,20,19)(7,21,25,24)(8,23,22,28)(10,27,29,11)(14,31,26,33)(15,32,35,34)(17,18,30,36)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#498866, #72213794, #72213801, … |
| maximal normal subgroup of | C2×#498866, #72213794, #72213801, … |
| maximal quotient of | #72213525, #72213530, C2×#498866 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406062);