Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15442 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,16,24,17,25,14,8)(4,18,20,9,21,27,22,28)(15,30,19,32,31,23,29,26);(1,3,5,14,7,17,6,16)(2,8,10,25,12,24,11,13)(4,19,20,15,21,29,22,31)(9,26,27,23,28,32,18,30);(3,14,17,16)(8,13,24,25)(15,29,31,19)(23,26,30,32);(1,4)(2,9)(3,15,17,31)(5,20)(6,22)(7,21)(8,23,24,30)(10,27)(11,18)(12,28)(13,26,25,32)(14,29,16,19)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915, #72287261, #72287308, … |
| maximal normal subgroup of | #72279915, #72287261, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338836);