Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512894 |
| 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 | |
1050 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1050 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:
(2,7)(3,12)(4,17)(5,19)(8,25)(9,24)(10,29)(11,20)(13,31)(14,15)(16,21)(23,27)(26,28)(30,32);(1,2)(3,8)(4,9)(5,20)(6,11)(7,19)(10,22)(12,23)(13,26)(14,30)(15,25)(16,28)(17,29)(18,24)(21,32)(27,31);(1,3)(2,8)(4,13)(5,21)(6,16)(7,23)(9,26)(10,30)(11,28)(12,19)(14,22)(15,17)(18,31)(20,32)(24,27)(25,29);(1,4,6,17)(2,9,11,29)(3,13,16,15)(5,18,19,22)(7,10,20,24)(8,26,28,25)(12,14,21,31)(23,30,32,27)Polycyclic presentation:
2^9|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,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 | #72283193, #72283253, #72283274, … |
| maximal normal subgroup of | #72283193, #72283253, #72283274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420092);