Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #407025 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14945 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 208 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:
(2,8)(3,12)(5,6)(10,25)(11,20)(13,23)(14,29)(15,16)(17,18)(21,26)(27,30)(28,32);(1,2)(3,13,16,23)(4,10)(5,20)(6,8)(7,11)(9,15,24,12)(14,28,30,32)(17,21)(18,25)(19,26)(22,29,31,27);(1,3)(2,9,11,24)(4,14)(5,12)(6,15)(7,16)(8,13,20,23)(10,22,26,31)(17,29)(18,27)(19,30)(21,28,25,32);(1,4)(2,10)(3,14)(5,18)(6,17)(7,19)(8,21)(9,22)(11,26)(12,27)(13,28)(15,29)(16,30)(20,25)(23,32)(24,31)Polycyclic presentation:
2^9|b^2=efh,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314221);