Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,18,11)(8,19,20,34)(9,14)(10,13)(12,21,25,35)(16,29)(17,28,32,24)(22,26)(23,31)(27,36)(30,33);(1,2,4,3)(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,15)(14,24)(19,33)(20,30)(21,31)(22,32)(25,27)(26,28)(29,34)(35,36);(5,8)(6,12,13,25)(7,16)(9,21)(10,20)(11,23,15,27)(14,19,26,34)(17,31)(18,30)(22,35)(24,33,28,29)(32,36);(1,3,4,2)(8,20)(12,25)(16,30)(19,34)(21,35)(23,27)(29,33)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=h,f^2=i,b^a=be,c^a=cf,e^a=ei,g^b=gi,d^c=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72302337, #72302371, #72302768, … |
| maximal normal subgroup of | #72302337, #72302371, #72302768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262074);