Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512898 |
| 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 |
| 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 | |
1037 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1037 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:
(2,9)(3,13)(5,18)(6,17)(11,25)(12,19)(14,32)(15,22)(16,21)(20,23)(24,28)(27,31);(1,2,8,12)(3,10,15,30)(4,16,7,11)(5,19,18,9)(6,21,17,25)(13,31,22,27)(14,29,23,26)(20,28,32,24);(1,3)(2,10)(4,14)(5,20)(6,22)(7,23)(8,15)(9,24)(11,26)(12,30)(13,17)(16,29)(18,32)(19,28)(21,27)(25,31);(1,4,8,7)(2,11,12,16)(3,14,15,23)(5,17,18,6)(9,25,19,21)(10,26,30,29)(13,32,22,20)(24,31,28,27)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,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 | #72306637, #72306778, #72306779, … |
| maximal normal subgroup of | #72306637, #72306778, #72306779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420097);