Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12698 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 32 points:
(1,2)(3,13,16,30)(4,11,6,22)(5,10)(7,12)(8,14,25,17)(9,18,27,20)(15,28,32,26)(19,24,31,23)(21,29);(1,3)(2,8,12,13)(4,14)(5,19)(6,16)(7,17)(9,26,27,23)(10,28,29,24)(11,30,22,25)(15,21)(18,32)(20,31);(1,4,7,6)(2,9)(3,14,17,16)(5,18,21,20)(8,23)(10,11)(12,27)(13,26)(15,31,19,32)(22,29)(24,30)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=hi,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fgh,g^b=gi,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72292696, C2×#152128 |
| maximal normal subgroup of | #72292696, C2×#152128 |
| maximal quotient of | C2×#152128, #72293068, #72293074, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59324);