Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395166 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,10)(7,9,11,12)(14,20)(15,23)(16,22)(17,18)(21,24)(27,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14,17,20,19,22,18,16)(15,21,25,27,23,28,26,24);(1,3,4,2)(7,11)(9,12)(13,15,17,25,19,23,18,26)(14,21,20,27,22,28,16,24);(13,16,18,22,19,20,17,14)(15,24,26,28,23,27,25,21)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72322439, #72322451, #72322459, … |
| maximal normal subgroup of | #72322439, #72322451, #72322459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320833);