Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,27)(19,29)(20,28)(22,30)(23,34)(25,31)(32,35)(33,36);(5,7)(6,12)(8,22)(9,19)(10,18)(11,20)(13,30)(14,28)(15,29)(16,26)(17,25)(21,27)(23,33)(24,32)(31,36)(34,35);(5,8,9,24,11,23,10,25)(6,13,14,21,15,31,26,34)(7,17,18,33,20,32,19,22)(12,27,16,30,29,35,28,36);(1,3,4,2)(8,23)(13,31)(17,32)(21,34)(22,33)(24,25)(27,35)(30,36)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280941, #72281031, #72281142 |
| maximal normal subgroup of | #72280941, #72281031, #72281142 |
| maximal quotient of | #72313431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266303);