Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14175 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14)(3,11,17,30)(4,18)(5,24,16,9)(6,13,7,12)(10,22)(15,25)(19,26)(20,27)(21,29)(23,28)(31,32);(1,3,6,16,8,17,7,5)(2,9,12,11,14,24,13,30)(4,19,21,15,23,20,22,31)(10,26,28,32,29,27,18,25);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,23)(9,25)(11,27)(12,18)(13,28)(14,29)(16,19)(17,31)(24,32)(26,30);(1,5,7,17,8,16,6,3)(2,11,13,9,14,30,12,24)(4,20,21,31,23,19,22,15)(10,27,28,25,29,26,18,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287261, #72287295, #72287348, … |
| maximal normal subgroup of | #72287261, #72287295, #72287348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283394);