Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14519 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,19,29,28,30)(4,17,22,24,23,32,21,10)(5,11,27,20,26,31,25,16);(1,3)(2,9)(4,20)(5,24)(6,28)(7,18)(8,19)(10,26)(11,21)(12,29)(13,15)(14,30)(16,23)(17,25)(22,31)(27,32);(1,4)(2,10,14,24)(3,16,19,20)(5,9)(6,22)(7,21)(8,23)(11,18,31,28)(12,32,13,17)(15,27)(25,29)(26,30);(1,5)(2,11)(3,17)(4,9)(6,27)(7,25)(8,26)(10,18)(12,16)(13,20)(14,31)(15,22)(19,32)(21,29)(23,30)(24,28)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284930, #72287286, #72287326, … |
| maximal normal subgroup of | #72284930, #72287286, #72287326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295583);