Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,11,16,26)(8,18,20,29)(9,14)(10,13)(12,21,27,22)(15,30)(17,25,32,36)(19,34)(23,28)(24,33)(31,35);(1,2)(3,4)(5,7)(6,11,13,26)(8,19,20,15)(9,17)(10,16)(12,24)(14,25,28,36)(18,30)(21,33,22,31)(23,32)(27,35)(29,34);(2,4)(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,27)(14,29)(16,19)(17,33)(18,28)(21,23)(25,34)(26,35)(30,36)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72262963, #72262971, #72262972, … |
| maximal normal subgroup of | #72262963, #72262971, #72262972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30831);