Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3192 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #24005 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,11)(4,15)(5,20)(6,9)(7,13)(8,19)(10,18)(12,27)(14,24)(16,23)(17,31)(21,28)(22,29)(25,32)(26,30);(7,22)(11,27)(23,26)(24,28);(1,3,5,21,6,12,18,14)(2,7,10,26,9,22,20,23)(4,16,19,29,17,30,32,13)(8,24,15,27,25,28,31,11);(4,17)(8,25)(11,27)(13,29)(15,31)(16,30)(19,32)(24,28)Polycyclic presentation:
2^9|c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fhi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219593, #72224559, #72224561, … |
| maximal normal subgroup of | #72219593, #72224559, #72224561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407303);