Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149927 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #25734 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,13,11,15,8,23)(7,17,21,36,20,12,28,31)(10,14,24,33,27,16,26,29)(18,35,19,32,25,30,22,34);(5,7,8,18)(6,12,29,31)(9,25,11,20)(10,22,26,28)(13,30,14,32)(15,17,33,36)(16,34,23,35)(19,24,21,27);(2,4)(5,8)(6,13)(7,18)(9,11)(10,24)(12,30)(14,29)(15,23)(16,33)(17,35)(19,28)(20,25)(21,22)(26,27)(31,32)(34,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72284670, #72284729, #72286052 |
| maximal normal subgroup of | #72284670, #72284729, #72286052 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59960);