Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4120 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)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,27,37)(16,30,34,39)(17,23,18,22)(20,29,36,28)(21,33,40,32)(26,38,31,35)(41,43,44,42);(2,4)(7,8)(10,11)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,37)(23,25)(24,36)(26,34)(30,41)(32,42)(33,43)(35,40)(39,44);(5,7,9,8)(6,10,12,11)(13,16,19,34)(14,21,24,40)(15,26,27,31)(17,32,18,33)(20,35,36,38)(22,30,23,39)(25,41,37,44)(28,42,29,43);(5,8,9,7)(6,11,12,10)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(6,12)(10,11)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,32,34,33)(20,25,36,37)(21,30,40,39)(26,42,31,43)(35,41,38,44)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,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,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394787);