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×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,20,18,24,17,23)(15,21,27,34,25,38,28,36)(16,29,32,40,33,22,30,35)(26,39,41,44,31,37,42,43);(5,6)(7,11)(8,10)(9,12)(14,20)(15,25)(16,30)(17,19)(21,36)(23,24)(26,41)(31,42)(32,33)(34,38)(35,40)(37,39);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,31)(17,28)(18,25)(19,27)(20,34)(22,39)(23,36)(24,38)(26,33)(29,37)(30,41)(32,42)(35,44)(40,43);(7,12)(9,11)(13,16,17,30,18,33,19,32)(14,22,23,40,24,29,20,35)(15,26,28,42,25,31,27,41)(21,37,36,44,38,39,34,43);(6,10)(9,11)(13,17,18,19)(14,23,24,20)(15,27,25,28)(16,32,33,30)(21,34,38,36)(22,35,29,40)(26,42,31,41)(37,44,39,43)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=f,e^2=i,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, 6972384);