Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232538 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1379 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1379 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,23,28)(19,29,32,40)(20,25,36,43)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(35,44,37,42);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,40)(27,29)(28,39)(30,35)(36,47)(37,46)(38,42)(43,45)(44,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,36)(18,25,28,43)(19,30,32,46)(21,35,22,37)(24,38,39,48)(26,42,27,44)(29,41,40,45)(31,34,33,47);(3,10)(6,13)(9,16)(12,14)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139736);