Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129125 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1776 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8286 |
| prime radical | C23 | #444 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 215 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 — 3 generators on 32 points:
(2,7,9,16,10,22,20,30)(3,11,14,28)(4,15)(8,23,25,21,12,29,26,24)(13,31,17,27)(19,32);(1,2)(3,12,14,8)(4,16)(5,20)(6,10)(7,19)(9,18)(11,26,28,25)(13,23,17,29)(15,30)(21,31,24,27)(22,32);(1,3,4,17,18,28,32,27,6,14,15,13,5,11,19,31)(2,8,16,23,9,26,22,24,10,12,30,29,20,25,7,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,c^b=ch,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214092, #72214308, #72214589, … |
| maximal normal subgroup of | #72214092, #72214308, #72214589, … |
| maximal quotient of | #72216227, C3×#129173, #72226302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36369);