Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512666 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #60817 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 871 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8)(5,10,20,9)(6,11)(13,27,28,21)(15,22,19,23)(16,29,30,32)(17,24)(18,25,31,26);(1,3)(2,7)(4,16)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(13,17)(14,20)(18,28)(24,32)(25,27)(26,29)(30,31);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,26)(10,25)(11,24)(12,27)(14,28)(15,16)(19,30)(20,31)(22,29)(23,32);(4,17)(8,24)(13,16)(18,31)(21,32)(25,26)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=f,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ehi,f^b=fgh,g^b=gi,d^c=di,e^c=ei,f^e=fhi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72306437, #72318931 |
| maximal normal subgroup of | #72306437, #72318931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419862);