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 | |
1098 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1098 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) = 128 |
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)(19,28)(20,33)(21,24)(22,27)(23,31)(25,34)(26,37)(29,40)(30,38)(32,39)(35,42)(36,41)(43,44)(45,47)(46,48);(6,12)(13,15)(17,19,22,32)(18,23,27,40)(20,30,36,46)(21,29,37,31)(24,28,26,39)(25,38,42,48)(33,43,41,47)(34,45,35,44);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(18,24)(19,29)(21,37)(25,33)(26,27)(28,39)(30,44)(31,32)(34,35)(41,42)(43,47)(45,46);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,20,22,36)(18,25,27,42)(19,30,32,46)(21,35,37,34)(23,38,40,48)(24,33,26,41)(28,43,39,47)(29,44,31,45)Polycyclic presentation:
2^9|a^2=hi,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, 151732);