Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18)(19,24)(20,23)(21,34)(22,25)(26,31)(27,32)(28,40)(29,35)(30,44)(33,36)(37,45)(38,43)(39,42)(41,47)(46,48);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(18,23)(20,31)(21,32)(24,35)(25,36)(26,27)(29,43)(30,42)(33,45)(34,37)(38,39)(40,47)(41,48)(44,46);(1,4,5,12,6,13,10,14)(2,7,9,16,3,11,8,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,35)(25,40)(26,38)(27,39)(31,43)(32,42)(33,41)(34,44)(36,47)(37,46)(45,48);(2,3)(7,11)(8,9)(15,16)(18,25)(19,28)(23,36)(26,37)(27,34)(29,41)(30,42)(43,48)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391178);