Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1039 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)(7,17,21,32)(8,14,9,23)(10,15)(11,16)(12,19)(13,22,30,24)(18,28)(20,26,35,36)(25,29)(27,34)(31,33);(1,2,3,4)(5,7)(6,12,16,28)(8,18)(9,19)(10,20)(11,21)(13,26,30,36)(14,17,23,32)(15,31,29,27)(22,33)(24,34)(25,35);(5,8)(6,13)(7,18)(9,11)(10,24)(12,26)(14,29)(15,23)(16,30)(17,27)(19,21)(20,34)(22,25)(28,36)(31,32)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=eh,f^b=fhi,d^c=dhi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286484, #72286492, #72286516, … |
| maximal normal subgroup of | #72286484, #72286492, #72286516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53841);