Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3152 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)(7,11)(10,12)(13,14,19,24)(15,25,28,36)(16,30,33,37)(17,23,18,22)(20,29,35,27)(21,32,38,31)(26,40,42,34)(39,43,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,29)(18,27)(19,28)(21,34)(22,36)(23,25)(24,35)(30,44)(31,43)(32,41)(33,42)(37,39)(38,40);(6,9)(10,12)(13,16,19,33)(14,21,24,38)(15,26,28,42)(17,31,18,32)(20,34,35,40)(22,30,23,37)(25,39,36,44)(27,41,29,43);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,25,35,36)(21,30,38,37)(26,41,42,43)(34,39,40,44);(5,7,8,11)(6,10,9,12)(15,28)(20,35)(25,36)(26,42)(27,29)(34,40)(39,44)(41,43)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,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, 6398289);