Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
6676 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)(5,6,9,10)(7,8,11,12)(14,19)(20,30)(21,31)(22,32)(33,41)(34,42)(35,36)(43,44);(6,10)(7,11)(14,19)(15,23)(21,31)(22,32)(24,37)(25,38)(35,36)(39,40);(5,7)(6,8)(9,11)(10,12)(13,14,18,19)(15,20,23,30)(16,21,27,31)(17,22,29,32)(24,33,37,41)(25,34,38,42)(26,35,28,36)(39,43,40,44);(5,8)(6,7)(9,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,30)(21,33)(22,34)(26,39)(27,37)(28,40)(29,38)(31,41)(32,42)(35,43)(36,44);(1,3,4,2)(13,16,18,27)(14,21,19,31)(15,24,23,37)(17,28,29,26)(20,33,30,41)(22,36,32,35)(25,40,38,39)(34,44,42,43);(13,17,18,29)(14,22,19,32)(15,25,23,38)(16,26,27,28)(20,34,30,42)(21,35,31,36)(24,39,37,40)(33,43,41,44)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534700);