Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493653 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63045 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2300 conjugacy classes of subgroups; only the 304 normal subgroups are shown.
This group has 2300 conjugacy classes of subgroups — too many to draw. Only its 304 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)(3,9,15,29)(4,10,16,7)(5,11)(6,12)(8,14,27,19)(13,28)(17,23)(18,30,20,24)(21,25)(22,32)(26,31);(2,7)(4,16)(8,22)(9,23)(10,12)(11,30)(13,17)(18,20)(24,25)(26,27)(28,29)(31,32);(1,3)(2,8)(4,17)(5,19)(6,15)(7,22)(9,25)(10,26)(11,29)(12,27)(13,16)(14,21)(18,31)(20,32)(23,24)(28,30);(1,3)(2,9)(4,13)(5,14)(6,15)(7,23)(8,25)(10,28)(11,27)(12,29)(16,17)(18,31)(19,21)(20,32)(22,24)(26,30)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72261708 |
| maximal normal subgroup of | #72261708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400849);