Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,10,12,15)(7,13)(11,18,19,17)(14,20)(21,22,25,30,26,28,24,29)(23,31,34,27,32,36,33,35);(1,2,3,4)(5,6)(7,11)(8,15)(9,12)(10,16)(13,19)(14,18)(17,20)(21,23)(22,27)(24,34)(25,33)(26,32)(28,35)(29,36)(30,31);(5,7,8,14,9,13,16,20)(6,11,15,18,12,19,10,17)(22,28)(27,35)(29,30)(31,36);(23,32)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,c^a=ch,d^b=di,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224967, #72225095, #72225106, … |
| maximal normal subgroup of | #72224967, #72225095, #72225106, … |
| maximal quotient of | #72224955, #72286564, C2×#183474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90670);