Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470149 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,20,25,34)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,40,44)(26,35,30,36)(33,41,43,42);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,24)(17,30)(18,26)(19,25)(21,33)(22,36)(23,34)(27,39)(28,42)(29,40)(31,35)(32,41)(37,44)(38,43);(1,3,4,2)(7,11)(9,12)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,43)(35,39)(36,44);(7,11)(9,12)(15,25)(20,34)(24,40)(26,30)(33,43)(35,36)(39,44)(41,42);(6,10)(9,12)(13,17,19,18)(14,22,23,31)(15,26,25,30)(16,28,29,32)(20,35,34,36)(21,37,38,27)(24,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377940);