Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C24 | C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(9,10,11,12)(13,14,19,22)(15,18,21,17)(16,24,27,23)(20,25,28,26);(2,4)(5,6)(7,8)(13,15,18,14,19,21,17,22)(16,23,25,20,27,24,26,28);(6,8)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,24)(22,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72215705, #72215970, #72215994, … |
| maximal normal subgroup of | #72215705, #72215970, #72215994, … |
| maximal quotient of | #72215644, #72215673, #72238287 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30763);