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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1681 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)(5,6,9,16,11,14,26,32)(7,18,21,33,19,12,25,30)(8,15,24,17,23,13,10,27)(20,28,22,34,35,29,36,31);(2,4)(5,7,10,22,11,19,24,36)(6,12,17,31,14,18,27,34)(8,20,9,25,23,35,26,21)(13,28,16,30,15,29,32,33);(6,13)(8,23)(9,26)(12,28)(14,15)(16,17)(18,29)(20,35)(21,25)(27,32)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,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.
| maximal subgroup of | #72312095, #72312111, #72312115, … |
| maximal normal subgroup of | #72312095, #72312111, #72312115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53787);