Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15)(7,16)(8,13,22,30)(9,14,24,31)(11,23)(12,19)(17,28)(18,26)(20,33)(21,29,35,36)(25,34)(27,32);(1,2,3,4)(5,7)(6,11)(8,20)(9,23)(10,19)(12,14)(13,28)(15,26)(16,31)(17,21)(18,24)(22,34)(25,29)(27,30)(32,35)(33,36);(5,7)(6,12)(8,17)(9,18)(10,19)(11,14)(13,27)(15,16)(20,21)(22,32)(23,24)(25,29)(26,31)(28,30)(33,36)(34,35);(5,8,9,21,10,22,24,35)(6,13,14,29,15,30,31,36)(7,17,18,20,19,32,23,34)(11,25,16,28,26,33,12,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72227326, #72312158, #72312207, … |
| maximal normal subgroup of | #72227326, #72312158, #72312207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263426);