Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512866 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| prime radical | #61110 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1042 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1042 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13)(3,14,18,32)(4,12,6,10)(5,11,21,30)(8,23,25,16)(9,26,28,17)(15,24,19,27)(20,31,22,29);(1,3)(2,8)(4,19)(5,16)(6,15)(7,18)(9,12)(10,28)(11,14)(13,25)(17,20)(21,23)(22,26)(24,29)(27,31)(30,32);(1,3)(2,9)(4,15)(5,17)(6,19)(7,18)(8,12)(10,25)(11,14)(13,28)(16,20)(21,26)(22,23)(24,31)(27,29)(30,32);(1,4,7,6)(2,10,13,12)(3,15,18,19)(5,20,21,22)(8,9,25,28)(11,29,30,31)(14,27,32,24)(16,26,23,17)Polycyclic presentation:
2^9|a^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,d^b=di,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72279417, #72286310, #72286325, … |
| maximal normal subgroup of | #72279417, #72286310, #72286325, … |
| maximal quotient of | #72279429, #72340621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420064);