Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,8,10,9,7,12,11)(13,14,17,23,33,42,31,20,19,25,16,22,18,24,34,38)(15,26,28,44,36,41,32,40,30,37,27,39,29,43,35,21);(1,2,3,4)(5,7)(6,9)(8,10)(11,12)(14,20)(16,31)(17,34)(18,33)(21,37)(22,24)(23,42)(25,38)(26,40)(27,32)(28,35)(29,36)(39,43)(41,44);(5,6)(7,9)(8,10)(11,12)(13,15,19,30)(14,21,25,40)(16,32,17,35)(18,36,33,29)(20,37,38,26)(22,41,23,43)(24,44,42,39)(27,34,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44578);