Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2733 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16680 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,16)(5,8)(6,11)(7,12)(10,18)(13,29)(14,24)(15,26)(17,32)(19,23)(20,22)(21,28)(25,31)(27,30);(2,8,12,11)(9,24,26,22)(10,23,28,16)(25,29,32,27);(1,3)(2,9)(4,17)(5,20)(6,14)(7,15)(8,22)(10,27)(11,24)(12,26)(13,18)(16,25)(19,31)(21,30)(23,32)(28,29);(1,4,5,18,7,19,6,21)(2,10,8,23,12,28,11,16)(3,13,14,17,15,30,20,31)(9,25,24,29,26,32,22,27)Polycyclic presentation:
2^9|b^2=g,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279455, #72284943, #72287267, … |
| maximal normal subgroup of | #72279455, #72284943, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377481);