Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153129 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
557 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 557 conjugacy classes of subgroups — too many to draw. Only its 52 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)(5,6,11,7)(8,12,22,15)(9,23,24,31)(10,17,27,14)(13,25,16,26)(18,33,29,36)(19,30,21,28)(20,35,32,34);(5,7,11,6)(8,18,22,29)(9,16,24,13)(10,14,27,17)(12,20,15,32)(19,35,21,34)(23,25,31,26)(28,36,30,33);(2,4)(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,29)(14,28)(16,18)(17,30)(21,27)(23,35)(24,32)(25,36)(26,33)(31,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60358);