Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133904 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2310 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
702 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 702 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,4,6)(5,8)(9,10,13,18,14,19,28,38)(11,20,23,39,22,16,32,33)(12,17,25,34,26,35,27,37)(15,29,31,21,30,40,36,24);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,23)(14,22)(16,17)(18,36)(19,30)(20,35)(21,26)(25,29)(27,40)(28,32)(31,38)(33,37)(34,39);(1,4,5,3)(2,7,8,6)(9,11)(10,16,19,20)(12,24)(13,23)(14,22)(15,17,30,35)(18,33,38,39)(21,26)(25,29)(27,40)(28,32)(31,34,36,37)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41190);