Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501114 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1097 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1097 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,18)(5,10,21,28)(6,11)(7,12)(8,16,25,20)(9,19,27,22)(14,30)(15,24)(17,29)(23,31,26,32);(1,3,7,17)(2,8,12,25)(4,15,6,14)(5,20,21,16)(9,24,27,30)(10,29,28,13)(11,23,18,26)(19,31,22,32);(3,14)(4,6)(5,21)(8,23)(9,27)(11,18)(13,30)(15,17)(16,32)(20,31)(24,29)(25,26);(1,4,7,6)(2,9,12,27)(3,15,17,14)(5,22,21,19)(8,24,25,30)(10,11,28,18)(13,26,29,23)(16,31,20,32)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237523, #72237553, #72312286, … |
| maximal normal subgroup of | #72237523, #72237553, #72312286, … |
| maximal quotient of | #72312047, #72312053, #72337467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408313);