Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,10,11,9,7,8,12)(13,14,18,25,20,27,17,22,21,28,16,24,19,26,36,39)(15,29,32,23,34,38,37,42,35,44,30,43,33,40,31,41);(1,2,3,4)(5,7,9,6)(8,12,10,11)(14,22)(16,17)(18,36)(19,20)(23,38)(24,26)(25,27)(28,39)(29,42)(30,37)(31,32)(33,34)(40,43)(41,44);(5,8,9,10)(6,11,7,12)(13,15)(14,23)(16,31)(17,32)(18,37)(19,34)(20,33)(21,35)(22,38)(24,29)(25,44)(26,42)(27,41)(28,43)(30,36)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=gh,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44124);