Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1102 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 144 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:
(1,2,5,8)(3,6,10,12)(4,7,11,14)(9,15,16,13)(17,18)(19,30)(20,34)(21,33)(22,41)(23,28)(24,29)(25,32)(26,38)(27,40)(31,36)(35,47)(37,46)(39,45)(42,44)(43,48);(6,12)(13,15)(17,19,24,27)(18,25,29,21)(20,35,39,48)(22,33,23,32)(26,43,46,47)(28,40,41,30)(31,38,44,37)(34,42,45,36);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20)(18,26)(19,31)(21,36)(22,37)(23,38)(24,39)(25,42)(27,44)(28,45)(29,46)(30,47)(32,35)(33,48)(34,41)(40,43);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,21,24,25)(18,27,29,19)(20,36,39,42)(22,30,23,40)(26,44,46,31)(28,32,41,33)(34,48,45,35)(37,47,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151731);