Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3485 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1063 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,6)(4,7)(5,8)(9,15)(10,12)(11,14)(13,16)(17,18,24,29)(19,30,35,49)(20,36,41,53)(21,42,45,55)(22,46,23,28)(25,34,50,33)(26,39,54,40)(27,44,56,43)(31,59,61,47)(32,48,63,74)(37,57,65,51)(38,68,71,52)(58,64,73,62)(60,77,80,75)(66,70,76,69)(67,72,79,78);(6,12)(13,15)(17,19)(18,25)(20,37)(21,32)(22,34)(23,33)(24,35)(26,51)(27,48)(28,49)(29,50)(30,46)(31,39)(36,47)(38,67)(40,61)(41,65)(42,58)(43,64)(44,62)(45,63)(52,75)(53,59)(54,57)(55,73)(56,74)(60,70)(66,72)(68,77)(69,80)(71,79)(76,78);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20,24,41)(18,26,29,54)(19,31,35,61)(21,38,45,71)(22,39,23,40)(25,47,50,59)(27,52,56,68)(28,53,46,36)(30,57,49,51)(32,60,63,80)(33,37,34,65)(42,66,55,76)(43,69,44,70)(48,72,74,78)(58,77,73,75)(62,79,64,67);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,21)(18,27)(19,32)(20,38)(22,44)(23,43)(24,45)(25,48)(26,52)(28,55)(29,56)(30,58)(31,60)(33,64)(34,62)(35,63)(36,66)(37,67)(39,70)(40,69)(41,71)(42,46)(47,72)(49,73)(50,74)(51,75)(53,76)(54,68)(57,77)(59,78)(61,80)(65,79)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151722);