Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395587 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2411 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14725 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,27)(3,14,18,29,19,32,17,9)(4,16,22,30,23,31,21,10)(5,24,25,20,26,11,28,15);(1,3,6,17,8,19,7,18)(2,9,12,14,13,29,27,32)(4,20,21,24,23,15,22,11)(5,16,25,30,26,31,28,10);(1,4)(2,10)(3,15)(5,9)(6,22)(7,21)(8,23)(11,18)(12,31)(13,30)(14,28)(16,27)(17,24)(19,20)(25,32)(26,29);(1,5)(2,11)(3,16)(4,9)(6,25)(7,28)(8,26)(10,18)(12,20)(13,24)(14,21)(15,27)(17,30)(19,31)(22,32)(23,29)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215938, #72219688, #72219691, … |
| maximal normal subgroup of | #72215938, #72219688, #72219691, … |
| maximal quotient of | C2×#395587 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302783);