Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146575 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10735 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1743 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1743 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 3 generators on 32 points:
(1,2,5,9)(3,11,14,29)(4,15,18,26)(6,10,21,27)(7,22,24,12)(8,20,25,17)(13,28,19,31)(16,23,30,32);(1,3)(2,7,10,23)(4,16,17,12)(5,19,21,14)(6,13)(8,11)(9,24)(15,28)(18,22)(20,30)(25,29,26,31)(27,32);(4,17)(8,15)(11,28)(12,16)(18,20)(22,30)(25,26)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^d=fi,g^d=gi,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72210444, #72210522, #72210534, … |
| maximal normal subgroup of | #72210444, #72210522, #72210534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53771);