Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,12)(10,15)(11,16)(13,20)(18,25)(19,31)(21,30)(22,29)(23,26)(24,28)(27,33)(32,35)(34,36);(6,12,16,15)(7,18)(8,21,22,23)(13,17,28,25)(14,29)(19,33,32,34)(20,24)(26,30);(5,7)(6,13)(8,19)(9,20)(10,24)(11,18)(12,25)(14,27)(15,17)(16,28)(21,34)(22,32)(23,33)(26,35)(29,36)(30,31);(1,3,4,2)(5,8)(6,14)(7,19)(9,23)(10,21)(11,22)(12,26)(13,27)(15,30)(16,29)(17,31)(18,32)(20,33)(24,34)(25,35)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,d^b=de,f^c=fi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224693, #72312202, #72312257, … |
| maximal normal subgroup of | #72224693, #72312202, #72312257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263693);