Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,15,12,16,10,28)(7,17,21,13,18,33,29,31)(8,22,24,27,25,14,30,9)(19,34,32,26,36,35,23,20);(5,7)(6,13)(8,23)(9,26)(10,29)(11,21)(12,18)(14,34)(15,33)(16,31)(17,28)(19,24)(20,27)(22,35)(25,32)(30,36);(2,4)(7,18)(13,31)(17,33)(19,36)(20,26)(21,29)(23,32)(34,35);(5,8)(6,14)(7,19)(9,15)(10,30)(11,24)(12,25)(13,20)(16,22)(17,35)(18,36)(21,32)(23,29)(26,31)(27,28)(33,34);(5,9,11,22,12,27,10,14)(6,8,15,24,16,25,28,30)(7,20,21,34,18,26,29,35)(13,32,33,36,31,23,17,19)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2955, #72286191, #72286234, … |
| maximal normal subgroup of | D4×#2955, #72286191, #72286234, … |
| maximal quotient of | #72286234, #72287015, #72287475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291056);