Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14311 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1197 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)(3,13,16,27)(4,17,20,25)(5,10)(6,11)(7,12)(8,15,23,21)(9,19,26,18)(14,22)(24,30)(28,31)(29,32);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,27)(11,13)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,22)(10,17)(11,25)(12,26)(13,28)(15,31)(16,30)(21,29)(23,24)(27,32);(1,2)(3,8)(4,17,20,25)(5,10)(6,11)(7,12)(9,19,26,18)(13,15)(14,28,30,32)(16,23)(21,27)(22,31,24,29)Polycyclic presentation:
2^9|a^2=h,d^2=efh,e^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284930, #72287274, #72287345, … |
| maximal normal subgroup of | #72284930, #72287274, #72287345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289085);