Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(2,4)(5,7,11,6,9,12,8,10);(6,10)(7,12)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224713, #72224720, #72224721 |
| maximal normal subgroup of | #72224713, #72224720, #72224721 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113299);