Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C4×B(4,𝔽2) |
| prime radical | #33567 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 190 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:
(1,2,7,13,8,14,6,22)(3,12,16,10,5,9,21,24)(4,11,19,27,20,28,18,31)(15,26,29,25,17,23,30,32);(2,9)(3,5)(10,22)(11,23)(12,14)(13,24)(15,17)(16,21)(25,31)(26,28)(27,32)(29,30);(1,3)(2,10)(4,15)(5,8)(6,21)(7,16)(9,13)(11,25)(12,22)(14,24)(17,20)(18,30)(19,29)(23,27)(26,31)(28,32);(1,4,7,19,8,20,6,18)(2,11,13,27,14,28,22,31)(3,15,16,29,5,17,21,30)(9,23,24,32,12,26,10,25)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317420, #72343000, #72343179, … |
| maximal normal subgroup of | #72317420, #72343000, #72343179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402965);