Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12155 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
701 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 701 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) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,8,10,9,12,7,11)(13,14)(15,25)(16,30,33,38)(17,23,18,22)(19,24)(20,27,36,28)(21,32)(26,37,31,35)(29,40)(34,39)(41,42)(43,44);(2,4)(6,10)(7,8)(11,12)(13,15,19,29)(14,20,24,36)(16,31,33,26)(17,27,18,28)(21,37,39,35)(22,25,23,40)(30,41,38,44)(32,43,34,42);(13,16)(14,21,24,39)(15,26)(17,34)(18,32)(19,33)(20,35,36,37)(22,38,23,30)(25,41,40,44)(27,43)(28,42)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38574);