Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1652 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)(5,6,10,15,11,16,9,14)(7,13,17,28,19,29,22,31)(8,12,20,25,21,26,23,30)(18,24,33,35,34,36,32,27);(6,12)(7,17)(9,10)(13,27)(14,30)(15,25)(16,26)(18,32)(19,22)(20,23)(24,28)(29,35)(31,36)(33,34);(2,4)(5,7,11,19)(6,13,16,29)(8,18,21,34)(9,22,10,17)(12,24,26,36)(14,31,15,28)(20,33,23,32)(25,35,30,27)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54786);