Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×Q8 | C22×C8 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C2×#555 | C4 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,21,25,36)(12,18,13,27)(14,19)(15,20)(16,23)(17,26,35,28)(22,32)(24,30,40,33)(29,34)(31,39)(37,38);(9,11)(10,16)(12,23)(13,22)(14,24)(15,25)(17,33)(18,36)(19,37)(20,32)(21,27)(26,39)(28,38)(29,40)(30,35)(31,34);(9,12)(10,17)(11,22)(13,15)(14,28)(16,30)(18,34)(19,27)(20,35)(21,31)(23,25)(24,39)(26,29)(32,33)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,e^b=eg,f^b=fgi,d^c=dgi,e^c=ei,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36784);