Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,15,13,17,9,25)(7,18,21,24,22,29,19,8)(10,16,28,34,12,30,26,35)(14,31,33,23,20,36,32,27);(2,4)(5,7,12,14,13,22,10,20)(6,8,16,23,17,24,30,27)(9,21,26,32,11,19,28,33)(15,18,35,31,25,29,34,36);(5,8,13,24)(6,14,17,20)(7,16,22,30)(9,18,11,29)(10,27,12,23)(15,32,25,33)(19,34,21,35)(26,31,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,e^c=ei,f^c=fi,f^d=fi,g^d=gi,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53788);