Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1145 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1145 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9)(3,10,12,16)(4,7,14,8)(6,13,15,11)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(6,15)(10,16)(17,19,22,28)(18,23,25,31)(20,29,21,26)(24,32,30,27);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,32)(27,29)(28,31)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72338245, #72338246, #72338410, … |
| maximal normal subgroup of | #72338245, #72338246, #72338410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151831);