Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31681 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24285 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,17,10)(4,7)(5,9)(6,12)(8,11)(14,30,32,27)(15,23,24,16)(18,20)(19,25)(21,26,31,29)(22,28);(2,9)(3,14,17,32)(4,11)(5,7)(6,20)(10,26,13,29)(15,30,24,27)(16,21,23,31)(18,25)(19,22);(1,3)(2,10)(4,15)(5,16)(6,21)(7,23)(8,17)(9,24)(11,13)(12,29)(14,19)(18,30)(20,32)(22,31)(25,27)(26,28);(1,4,8,9)(2,7,11,5)(3,15,17,24)(6,18,22,25)(10,23,13,16)(12,20,28,19)(14,29,32,26)(21,30,31,27)Polycyclic presentation:
2^9|a^2=efh,b^2=efh,d^2=i,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#510518, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#510518, #72320520, … |
| maximal quotient of | C2×#510518, #72250949, #72257141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417714);