Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125098 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
478 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6)(7,9,11,12)(8,10)(13,14,18,25,20,26,19,24)(15,27,31,39,32,44,30,43)(16,33,35,22,36,23,17,37)(21,28,40,29,42,34,41,38);(5,7,8,11)(6,9)(10,12)(13,15,17,38,19,31,36,28,20,32,35,29,18,30,16,34)(14,21,33,27,25,41,22,43,26,42,23,44,24,40,37,39);(1,2,3,4)(13,16,18,35,20,36,19,17)(14,22,24,33,26,37,25,23)(15,28,30,38,32,34,31,29)(21,39,40,44,42,43,41,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,d^b=dh,f^b=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48114);