Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1024 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)(7,10)(8,9)(11,12)(14,18)(15,19)(16,22)(20,24)(21,28)(23,27)(25,26);(2,4)(13,14,19,26,16,21,23,24,17,20,27,28,22,25,15,18);(6,9)(10,12)(14,20)(18,24)(21,25)(26,28);(5,7,8,11)(6,10,9,12)(14,20)(18,24)(21,25)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=f,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285820, #72285913, #72285952, … |
| maximal normal subgroup of | #72285820, #72285913, #72285952, … |
| maximal quotient of | #72287519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402749);