Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,24,26,32)(13,27)(14,16)(15,20)(17,22,35,40)(18,29)(19,23)(25,37,28,38)(30,36,39,34)(31,33);(2,5)(6,8)(9,11,13,14,15,23,29,31)(10,16,18,21,20,33,27,19)(12,22,28,30,26,40,25,39)(17,32,36,38,35,24,34,37);(9,12)(10,17)(11,22)(13,28)(14,30)(15,26)(16,32)(18,36)(19,37)(20,35)(21,38)(23,40)(24,33)(25,29)(27,34)(31,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=ehi,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287310, #72287321, #72287358, … |
| maximal normal subgroup of | #72287310, #72287321, #72287358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402184);