Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150977 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11952 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,4,12,6,25,19,11,5,10,22,30,24,31,17,29,8,15,21,27,26,14,20,32,7,13,23,9,18,28,3,16);(2,9)(3,4)(5,7)(6,18)(10,27)(11,14)(12,13)(15,30)(16,28)(17,21)(19,23)(20,22)(24,26)(25,32)(29,31);(2,10,15,13)(3,17)(4,21)(9,27,30,12)(11,16,32,29)(14,28,25,31)(19,20)(22,23)Polycyclic presentation:
2^9|a^2=e,c^2=fh,d^2=gh,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214100, #72214589, #72214592, … |
| maximal normal subgroup of | #72214100, #72214589, #72214592, … |
| maximal quotient of | #72216259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58173);