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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 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)(13,14,20,17,19,23,18,24)(15,25,32,21,28,38,27,35)(16,22,33,29,30,37,31,36)(26,39,42,34,40,44,41,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,23,35)(16,26,30,40)(17,25,24,38)(18,32,20,27)(22,34,37,43)(29,39,36,44)(31,42,33,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,26)(17,29)(18,31)(19,30)(20,33)(21,34)(23,37)(24,36)(25,39)(27,41)(28,40)(32,42)(35,43)(38,44);(7,12)(9,11)(13,17,20,23,19,24,18,14)(15,21,32,38,28,35,27,25)(16,29,33,37,30,36,31,22)(26,34,42,44,40,43,41,39);(6,10)(9,11)(14,23)(16,30)(17,24)(21,35)(25,38)(26,40)(31,33)(41,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6972389);