Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512902 |
| 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 | C4⋊Q8 | C24 |
| prime radical | #61110 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1036 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)(3,14,18,31)(4,12,6,10)(5,21,24,11)(8,22,28,16)(9,23,26,17)(15,29,19,27)(20,30,25,32);(1,3)(2,8)(4,19)(5,22)(6,15)(7,18)(9,10)(11,31)(12,26)(13,28)(14,21)(16,24)(17,20)(23,25)(27,30)(29,32);(1,3)(2,9)(4,15)(5,23)(6,19)(7,18)(8,10)(11,31)(12,28)(13,26)(14,21)(16,20)(17,24)(22,25)(27,32)(29,30);(1,4,7,6)(2,10,13,12)(3,15,18,19)(5,20,24,25)(8,26,28,9)(11,30,21,32)(14,27,31,29)(16,17,22,23)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gh,d^b=di,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,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 | #72283237, #72283255, #72283294, … |
| maximal normal subgroup of | #72283237, #72283255, #72283294, … |
| maximal quotient of | #72340686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420100);