Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,35)(16,30,31,38)(17,22,18,23)(20,28,36,27)(21,33,37,32)(26,40,41,34)(39,43,44,42);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,29)(14,20,24,36)(16,26,31,41)(17,28,18,27)(21,34,37,40)(22,25,23,35)(30,44,38,39)(32,43,33,42);(7,11)(9,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,31)(20,34)(22,38)(23,30)(24,37)(25,39)(27,42)(28,43)(29,41)(35,44)(36,40);(7,11)(9,12)(16,31)(21,37)(26,41)(30,38)(32,33)(34,40)(39,44)(42,43);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,31,33)(20,35,36,25)(21,38,37,30)(26,42,41,43)(34,44,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377366);