Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14,18,23,19,15,17,21)(16,24,26,20,27,28,25,22);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,14)(16,20,27,22)(17,21,18,23)(24,25,28,26);(2,4)(5,8,9,7)(6,10,11,12)(13,14,19,15)(16,20,27,22)(17,23,18,21)(24,25,28,26);(13,16,17,25,19,27,18,26)(14,20,23,28,15,22,21,24)Polycyclic presentation:
2^9|a^2=ei,b^2=ef,c^2=ef,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321853, #72321874, #72321910, … |
| maximal normal subgroup of | #72321853, #72321874, #72321910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320699);