Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13436 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,12,8,21)(7,16,19,27,20,33,18,13)(9,23,22,30,24,15,26,31)(17,32,35,29,25,36,34,28);(6,12)(7,17)(9,24)(13,28)(16,32)(18,35)(19,34)(20,25)(22,26)(27,29)(30,31)(33,36);(2,4)(5,7)(6,13,12,27)(8,19,10,18)(9,25)(11,20)(14,16)(15,28,23,29)(17,24)(21,33)(22,35,26,34)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311842, #72311884, #72311887, … |
| maximal normal subgroup of | #72311842, #72311884, #72311887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55665);