Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
544 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9,11,6)(4,7,12,14)(5,8)(10,15)(13,16)(17,18)(19,27,30,23)(20,31,33,24)(21,25)(22,26)(28,37)(29,36,43,39)(32,44,45,38)(34,40)(35,42)(41,46)(47,48);(6,9)(10,16)(17,19,22,30)(18,23,26,27)(20,28,33,42)(21,29,34,43)(24,37,31,35)(25,39,40,36)(32,41,45,48)(38,46,44,47);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,20,22,33)(18,24,26,31)(19,28)(21,32,34,45)(23,35)(25,38,40,44)(27,37)(29,41)(30,42)(36,46)(39,47)(43,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,22,34)(18,25,26,40)(19,29,30,43)(20,32,33,45)(23,36,27,39)(24,38,31,44)(28,41,42,48)(35,46,37,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=efhi,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283465, #72283568, #72283882, … |
| maximal normal subgroup of | #72283465, #72283568, #72283882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253468);