Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13340 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2011 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
548 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)(7,16)(8,22)(9,14)(10,25)(11,15)(12,19)(13,17)(18,33)(20,31)(21,30)(23,36)(24,29)(26,32)(27,35)(28,34);(5,7,17,32,10,18,23,35,11,20,24,34,9,21,8,19)(6,12,22,30,14,28,29,31,15,27,36,33,25,26,13,16);(7,18,20,21)(8,23)(12,26,27,28)(13,29)(16,30,31,33)(17,24)(19,34,35,32)(22,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250573, #72250585, #72250618, … |
| maximal normal subgroup of | #72250573, #72250585, #72250618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56402);