Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
685 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 685 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,15,22,16,21,37,52,17,23,36,53,14,38,13,20)(11,24,28,45,27,51,26,42,29,55,35,18,40,43,62,44)(12,30,33,47,32,19,46,48,34,49,66,70,39,50,31,64)(25,56,59,69,60,72,57,41,61,67,65,63,58,54,71,68);(1,2)(3,5)(4,6)(7,8)(9,11,16,40,17,29,14,27)(10,18,21,42,23,45,38,44)(12,25,32,58,34,61,39,60)(13,35,15,26,37,28,36,62)(19,41,49,69,50,68,30,63)(20,51,22,24,52,43,53,55)(31,65,33,57,46,59,66,71)(47,56,48,54,70,67,64,72);(1,3)(2,5)(4,8)(6,7)(9,12)(10,19)(11,25)(13,31)(14,39)(15,33)(16,32)(17,34)(18,41)(20,47)(21,49)(22,48)(23,50)(24,54)(26,57)(27,60)(28,59)(29,61)(30,38)(35,65)(36,66)(37,46)(40,58)(42,69)(43,67)(44,63)(45,68)(51,56)(52,70)(53,64)(55,72)(62,71)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44342);