Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,13,10,14,21,28)(7,12,17,26,15,27,29,36)(8,11,20,24,19,23,30,34)(16,22,31,33,18,25,32,35);(1,2,3,4)(6,11)(7,15)(12,25)(13,24)(14,23)(16,18)(17,29)(22,27)(26,35)(28,34)(31,32)(33,36);(5,7,9,17,10,15,21,29)(6,12,13,26,14,27,28,36)(8,18,20,32,19,16,30,31)(11,22,24,33,23,25,34,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311919, #72311939, #72311946, … |
| maximal normal subgroup of | #72311919, #72311939, #72311946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54530);