Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#447 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
797 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 797 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18)(19,31)(20,36)(21,27)(22,42)(23,29)(24,30)(25,34)(26,39)(28,44)(32,60)(33,58)(35,59)(37,52)(38,69)(40,56)(41,55)(43,48)(45,54)(46,53)(47,74)(49,73)(50,65)(51,72)(57,63)(61,79)(62,66)(64,77)(67,71)(68,78)(70,76)(75,80);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,25)(20,32)(21,33)(22,43)(23,46)(24,35)(26,47)(27,48)(28,53)(29,49)(30,50)(31,42)(34,44)(36,66)(37,61)(38,70)(39,63)(40,64)(41,65)(45,73)(51,77)(52,76)(54,59)(55,58)(56,75)(57,67)(60,69)(62,71)(68,80)(72,74)(78,79);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20)(18,26)(19,32)(21,37)(22,38)(23,39)(24,40)(25,47)(27,36)(28,51)(29,56)(30,52)(31,57)(33,61)(34,62)(35,64)(41,68)(42,67)(43,70)(44,71)(45,72)(46,63)(48,66)(49,75)(50,76)(53,77)(54,78)(55,69)(58,60)(59,79)(65,80)(73,74);(17,21,24,23)(18,27,30,29)(19,33,35,46)(20,37,40,39)(22,41,45,44)(25,48,50,49)(26,36,52,56)(28,42,55,54)(31,58,59,53)(32,61,64,63)(34,43,65,73)(38,68,72,71)(47,66,76,75)(51,67,69,78)(57,60,79,77)(62,70,80,74)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391751);