Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #31077 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
571 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,16)(4,22)(5,28)(6,14)(7,11)(8,15)(9,18)(10,25)(12,33)(13,30)(17,39)(19,47)(20,38)(21,43)(23,57)(24,59)(26,40)(27,41)(29,45)(31,37)(32,42)(34,48)(35,50)(36,54)(44,60)(46,53)(49,51)(52,63)(55,64)(56,62)(58,61);(1,3,8,21)(2,9,15,38)(4,23,27,56)(5,20,7,18)(6,31,33,19)(10,39,42,63)(11,43,28,16)(12,46,14,36)(13,47,45,37)(17,34,52,26)(22,49,41,64)(24,55,60,51)(25,58,32,50)(29,53,30,54)(35,48,61,40)(44,62,59,57);(1,4)(2,10)(3,17)(5,26)(6,32)(7,34)(8,27)(9,35)(11,44)(12,41)(13,48)(14,22)(15,42)(16,49)(18,23)(19,55)(20,56)(21,52)(24,30)(25,33)(28,59)(29,60)(31,51)(36,62)(37,63)(38,61)(39,47)(40,45)(43,64)(46,57)(50,54)(53,58)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209895, #72209900 |
| maximal normal subgroup of | #72209895 |
| maximal quotient of | #72227840, #72227997, #72228035 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60324);