Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | C23 | #532 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
634 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,14,8)(3,11,4,7,12,15,13,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,5,4,6,12,14,13)(2,7,8,11,10,16,9,15);(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,4,5,12,6,13,14,3)(2,7,9,15,10,16,8,11)(19,26)(22,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225265, #72225270, #72225272, … |
| maximal normal subgroup of | #72225265, #72225270, #72225272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120471);