Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,14,19,23)(15,16,20,28)(17,21,31,37)(18,32,33,22)(24,26,29,35)(25,39,34,27)(30,42,43,36)(38,44,41,40);(6,10)(7,11)(13,15,19,20)(14,16,23,28)(17,29,31,24)(18,25,33,34)(21,35,37,26)(22,27,32,39)(30,41,43,38)(36,44,42,40);(5,7,9,11)(6,8,10,12)(13,14)(15,16)(17,21)(18,22)(19,23)(20,28)(24,26)(25,27)(29,35)(30,36)(31,37)(32,33)(34,39)(38,40)(41,44)(42,43);(5,8)(6,11)(7,10)(9,12)(13,16)(14,20)(15,23)(17,26)(18,27)(19,28)(21,29)(22,34)(24,37)(25,32)(30,40)(31,35)(33,39)(36,41)(38,42)(43,44);(1,3,4,2)(13,17)(14,21)(15,24)(16,26)(18,30)(19,31)(20,29)(22,36)(23,37)(25,38)(27,40)(28,35)(32,42)(33,43)(34,41)(39,44);(13,18,19,33)(14,22,23,32)(15,25,20,34)(16,27,28,39)(17,30,31,43)(21,36,37,42)(24,38,29,41)(26,40,35,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533762);