Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,14)(9,12)(10,28)(11,18)(13,22)(15,25)(16,26)(17,24)(20,31)(21,34)(23,30)(27,32)(29,35)(33,36);(1,2)(3,4)(6,12)(7,20)(8,24)(9,25)(10,26)(13,31)(14,16)(15,17)(18,28)(19,35)(21,23)(22,36)(30,32)(33,34);(2,4)(5,7)(6,13)(8,21)(9,27)(10,29)(11,23)(12,30)(14,32)(15,31)(16,34)(17,35)(18,33)(19,28)(20,24)(22,26)(25,36);(5,8,11,24)(6,14,18,17)(7,21,23,20)(9,25,10,26)(12,15,28,16)(13,32,33,35)(19,34,30,31)(22,27,36,29)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224936, #72224952, … |
| maximal normal subgroup of | #72224903, #72224936, #72224952, … |
| maximal quotient of | #72224896, #72237266, C2×#498880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406076);