Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1026 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,14)(8,16)(9,18)(10,13)(11,15)(12,19)(17,20)(21,22,26,30)(23,27,29,25)(24,31,33,35)(28,32,36,34);(5,7,10,8)(6,11,13,20)(9,17,19,15)(12,16,18,14)(21,23,26,29)(22,25,30,27)(24,28,33,36)(31,34,35,32);(2,4)(21,24,26,33)(22,28,30,36)(23,31,29,35)(25,32,27,34);(5,8,10,7)(6,11,13,20)(9,15,19,17)(12,16,18,14)(21,25,26,27)(22,29,30,23)(24,32,33,34)(28,35,36,31)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152258);