Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1704 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)(7,12)(8,18,19,11)(9,21,22,13)(10,14)(15,27,23,30)(16,24,32,33)(17,28)(20,25)(26,31)(29,34)(35,36);(6,11)(7,15)(9,22)(12,26)(13,29)(14,18)(16,31)(17,23)(20,34)(21,25)(24,27)(28,36)(30,33)(32,35);(2,4)(5,7)(6,12)(8,16)(9,23)(10,17)(11,24)(13,30)(14,28)(15,22)(18,33)(19,32)(20,35)(21,27)(25,36)(26,29)(31,34)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,f^c=fi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312095, #72312113, #72312136, … |
| maximal normal subgroup of | #72312095, #72312113, #72312136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54951);