Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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×#3326 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(15,21)(16,22)(17,23)(18,20)(19,24)(25,37)(26,36)(27,33)(28,40)(29,39)(30,38)(31,35)(32,34)(41,43)(42,44);(5,6)(7,11)(8,10)(9,12)(14,20)(15,25)(16,28)(17,18)(21,35)(22,38)(23,24)(26,41)(29,42)(30,32)(33,37)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,29)(17,31)(18,27)(19,25)(20,33)(22,39)(23,35)(24,37)(26,30)(28,41)(32,42)(34,44)(36,38)(40,43);(7,12)(9,11)(13,16)(14,22)(15,26)(17,28)(18,32)(19,30)(20,34)(21,36)(23,40)(24,38)(25,29)(27,41)(31,42)(33,43)(35,44)(37,39);(6,10)(9,11)(13,17,19,18)(14,20,24,23)(15,27,25,31)(16,28,30,32)(21,35,37,33)(22,34,38,40)(26,41,29,42)(36,44,39,43)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969808);