Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15501 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,16,24,17,8,15,25)(4,18,21,9,22,27,20,28)(14,30,19,23,31,26,32,29);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,25)(11,13)(12,24)(14,21)(18,29)(20,31)(22,32)(23,27)(28,30);(1,4,5,20,7,22,6,21)(2,9,10,27,11,28,12,18)(3,14,15,19,17,31,16,32)(8,23,25,30,13,29,24,26);(2,10,11,12)(8,24,13,25)(9,27,28,18)(23,26,29,30)Polycyclic presentation:
2^9|a^2=g,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284607, #72287261, #72287280, … |
| maximal normal subgroup of | #72284607, #72287261, #72287280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341611);