Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1333 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7178 |
| prime radical | C23 | #360 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,14,30,15)(3,16)(4,22,25,48,27,46,28,12)(6,31,32,10,34,43,26,44)(7,35,29,13,24,47,33,42)(9,23,37,56)(17,45,52,40)(18,49,54,39)(19,58)(20,59)(21,51)(36,60)(38,61)(41,53)(50,55)(57,64,62,63);(1,3,6,18,33,55,25,17,30,53,32,62,24,19,27,56,8,21,34,54,29,61,28,52,5,20,26,57,7,36,4,23)(2,9,12,38,47,64,31,16,11,45,46,58,42,39,44,59,14,37,48,50,35,63,43,51,15,40,22,60,13,49,10,41)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fgh,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72223653, #72223659, #72223685, … |
| maximal normal subgroup of | #72223653, #72223659, #72223685, … |
| maximal quotient of | #72223650, C2×#94056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1252);