Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
530 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)(7,11,16,24)(8,19)(9,14)(10,12)(13,20)(15,23,18,26)(17,32,31,25)(21,34)(22,27)(28,29)(30,36,33,35);(5,7,8,17,9,18,21,33,10,16,20,31,22,15,29,30)(6,11,13,25,14,26,28,36,12,24,19,32,27,23,34,35);(2,4)(7,15,16,18)(8,20)(11,23,24,26)(13,19)(17,33,31,30)(21,29)(25,36,32,35)(28,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239415, #72239418, #72239428, … |
| maximal normal subgroup of | #72239415, #72239418, #72239428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54324);