Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144394 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10336 |
| commutator, Frattini, cosocle | C22×#41 | C23 |
| prime radical | #23896 | C2 |
| LCS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 855 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,24)(7,12)(8,15)(9,20)(11,27)(14,42)(16,49)(17,41)(19,54)(21,37)(22,38)(23,35)(25,32)(26,34)(28,58)(29,51)(30,48)(31,57)(33,55)(36,47)(39,44)(40,52)(43,53)(45,63)(46,59)(50,62)(56,61)(60,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,28)(10,13)(11,36)(12,30)(18,53)(19,43)(20,44)(21,45)(22,47)(23,48)(24,29)(25,50)(26,51)(27,52)(31,39)(32,60)(33,62)(34,49)(35,41)(37,59)(38,40)(42,54)(46,56)(55,64)(57,58)(61,63);(1,4,7,21)(2,9,12,32)(3,14,17,45)(5,19,23,56)(6,25,27,20)(8,28,30,60)(10,33,35,31)(11,37,24,18)(13,39,41,62)(15,46,48,43)(16,50,52,44)(22,57,26,55)(29,53,36,59)(34,61,38,54)(40,63,49,42)(47,64,51,58)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,g^b=gi,d^c=dg,e^c=ei,e^d=ei,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72209847 |
| maximal normal subgroup of | #72209847 |
| maximal quotient of | #72227681, #72227720, #72227721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51591);