Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121958 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1445 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
544 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 48 points:
(1,2)(3,6,10,14)(4,11)(5,8)(7,12)(9,13,15,16)(17,18)(19,23,28,38)(20,30,32,45)(21,26)(22,25)(24,31,40,47)(27,42)(29,37,33,41)(34,35)(36,46)(39,44)(43,48);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19,21,33,22,28,34,29)(18,23,35,37,25,38,26,41)(20,27,31,46,32,43,47,44)(24,39,45,48,40,36,30,42);(1,4)(2,7,8,11)(3,9)(5,12)(6,13,14,16)(10,15)(17,20,21,31,22,32,34,47)(18,24,35,45,25,40,26,30)(19,27,33,46,28,43,29,44)(23,36,37,42,38,39,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,e^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,d^c=di,e^c=ei,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29154);