Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129633 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,13,24,29)(7,15,17,31,18,12,22,28)(8,19,21,30,20,11,23,26)(16,27,33,36,32,25,34,35);(1,2)(3,4)(6,11)(12,27)(13,19)(14,30)(15,25)(26,29)(28,35)(31,36);(2,4)(5,7,10,18)(6,12,13,15)(8,16,20,32)(9,22,24,17)(11,25,19,27)(14,31,29,28)(21,34,23,33)(26,36,30,35)Polycyclic presentation:
2^9|a^2=g,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=eh,g^b=gh,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303727, #72313159, #72313178, … |
| maximal normal subgroup of | #72303727, #72313159, #72313178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55014);