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 | |
1662 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)(8,18,20,11)(9,13)(10,14)(15,26)(16,32,33,23)(17,27)(19,28,34,24)(21,30)(22,29)(25,31,36,35);(6,11)(7,15)(9,21)(12,25)(13,28)(14,18)(16,31)(17,22)(19,34)(23,26)(24,30)(27,36)(29,32)(33,35);(2,4)(5,7,10,17)(6,12,14,27)(8,16,20,33)(9,22,21,15)(11,23,18,32)(13,29,30,26)(19,35,34,31)(24,36,28,25)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^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.
| maximal subgroup of | #72312834, #72313143, #72337433, … |
| maximal normal subgroup of | #72312834, #72313143, #72337433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54782);