Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #268211 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
994 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,17,10,8,11,15)(7,13,18,20,9,14,16,19)(21,22)(23,30,32,35)(24,28)(25,29)(26,31,27,33)(34,36);(1,2)(3,4)(5,7,10,9)(6,13,8,14)(11,18,12,16)(15,20,17,19)(21,23)(22,26)(24,33)(25,32)(27,29)(28,35)(30,36)(31,34);(2,4)(5,8,10,6)(7,14,9,13)(11,15,12,17)(16,19,18,20)(24,34)(28,36)(30,35)(31,33);(5,9,10,7)(6,14,8,13)(11,16,12,18)(15,19,17,20)(21,23)(22,27)(24,33)(25,32)(26,29)(28,30)(31,34)(35,36)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=h,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fh,e^d=ei,f^d=fhi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 175422);