Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,35,19,25,18,24)(15,26,17,21,29,37,28,23)(16,30,32,40,34,22,33,39)(27,42,31,38,44,41,43,36);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,25,23)(16,27,34,44)(17,20,28,18)(22,36,30,38)(24,37,35,26)(31,32,43,33)(39,41,40,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,27)(17,31)(18,33)(19,34)(20,32)(21,36)(23,38)(24,40)(25,30)(26,41)(28,43)(29,44)(35,39)(37,42);(7,12)(9,11)(13,17,18,15,19,28,20,29)(14,23,24,37,25,21,35,26)(16,31,33,27,34,43,32,44)(22,38,40,42,30,36,39,41);(6,10)(9,11)(13,18,19,20)(14,24,25,35)(15,17,29,28)(16,32,34,33)(21,37,23,26)(22,39,30,40)(27,43,44,31)(36,41,38,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972244);