Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128808 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #440 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
262 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,6,10,7,12,5,11)(3,13,14,8,15,4,9,16)(17,18,21,26)(19,29)(20,35,37,27)(22,40,38,25)(23,41,39,28)(24,36)(30,43)(31,45)(32,44)(33,48)(34,46)(42,47);(1,3,5,9,7,15,6,14)(2,8,11,13,12,16,10,4)(17,19,22,32,39,30,37,31,21,34,38,47,23,33,20,36)(18,24,27,43,41,42,40,29,26,45,35,48,28,44,25,46);(1,4,6,16,7,13,5,8)(2,9,10,3,12,14,11,15)(19,30,34,33)(20,37)(22,38)(24,42,45,44)(25,40)(27,35)(29,43,46,48)(31,32,36,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ei,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,d^c=di,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285527, #72285993, #72287115, … |
| maximal normal subgroup of | #72285527, #72285993, #72287115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36212);