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)(13,14,18,23,19,22,32,17)(15,24,26,35,27,38,33,20)(16,28,31,29,30,21,36,37)(25,40,42,34,41,39,44,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,22,35)(16,25,30,41)(17,24,23,38)(18,33,32,26)(21,34,28,43)(29,39,37,40)(31,44,36,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,34)(22,28)(23,37)(24,39)(26,42)(27,41)(32,36)(33,44)(35,43)(38,40);(7,12)(9,11)(13,17,32,22,19,23,18,14)(15,20,33,38,27,35,26,24)(16,29,36,28,30,37,31,21)(25,34,44,40,41,43,42,39);(6,10)(9,11)(14,22)(16,30)(17,23)(20,35)(24,38)(25,41)(31,36)(42,44)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,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, 6972133);