Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489739 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,12)(4,15)(5,11)(6,10)(7,9)(8,13)(14,16)(17,18,24,30)(19,31,35,61)(20,26,38,53)(21,29,23,27)(22,41,45,55)(25,47,50,34)(28,40,58,44)(32,62,65,60)(33,67,46,56)(36,54,39,51)(37,70,43,73)(42,59,75,49)(48,68,77,64)(52,72,57,69)(63,79,66,80)(71,76,74,78);(1,3,7,4)(2,8,9,16)(5,14,6,13)(10,15,11,12)(17,19)(18,25)(20,32)(21,33)(22,42)(23,46)(24,35)(26,48)(27,49)(28,56)(29,59)(30,50)(31,41)(34,44)(36,63)(37,71)(38,65)(39,66)(40,47)(43,74)(45,75)(51,76)(52,79)(53,77)(54,78)(55,61)(57,80)(58,67)(60,73)(62,70)(64,69)(68,72);(1,4,7,3)(2,8,9,16)(5,13,6,14)(10,15,11,12)(17,20,24,38)(18,26,30,53)(19,32,35,65)(21,39,23,36)(22,43,45,37)(25,48,50,77)(27,54,29,51)(28,57,58,52)(31,60,61,62)(33,66,46,63)(34,68,47,64)(40,69,44,72)(41,73,55,70)(42,74,75,71)(49,78,59,76)(56,80,67,79);(2,9)(8,16)(10,11)(12,15)(17,21,24,23)(18,27,30,29)(19,33,35,46)(20,36,38,39)(22,40,45,44)(25,49,50,59)(26,51,53,54)(28,55,58,41)(31,56,61,67)(32,63,65,66)(34,42,47,75)(37,69,43,72)(48,76,77,78)(52,70,57,73)(60,79,62,80)(64,74,68,71)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=i,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397120);