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 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1044 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1044 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:
(2,8)(3,12)(4,15)(5,18)(10,16)(11,29)(13,32)(14,20)(17,22)(19,28)(23,26)(25,31);(1,2)(3,9)(4,16)(5,19)(6,11)(7,10)(8,15)(12,31)(13,27)(14,24)(17,26)(18,29)(20,25)(21,28)(22,30)(23,32);(1,3)(2,9)(4,17)(5,20)(6,22)(7,14)(8,23)(10,24)(11,30)(12,18)(13,21)(15,32)(16,26)(19,25)(27,28)(29,31);(2,10)(8,16)(9,24)(11,28)(19,29)(23,26)(25,31)(27,30)Polycyclic presentation:
2^9|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 | #72306810, #72306817, #72306840 |
| maximal normal subgroup of | #72306810, #72306817, #72306840 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420094);