Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396100 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2414 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14730 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1392 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)(3,13,15,8)(4,9)(5,12)(6,10)(7,11)(14,22,30,29)(16,24,20,23)(17,32,31,25)(18,26)(19,27)(21,28);(1,3)(2,8)(4,17)(5,20)(6,16)(7,15)(9,25)(10,23)(11,13)(12,24)(14,18)(19,31)(21,30)(22,28)(26,29)(27,32);(1,4)(2,9)(3,14)(5,21)(6,18)(7,19)(8,22)(10,26)(11,27)(12,28)(13,29)(15,30)(16,31)(17,20)(23,25)(24,32);(2,10,11,12)(8,23,13,24)(9,26,27,28)(22,25,29,32)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^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,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287261, #72287295, #72287341, … |
| maximal normal subgroup of | #72287261, #72287295, #72287341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303296);