Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2746 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)(7,12)(8,10)(9,11)(13,14)(15,23)(16,28)(17,21)(18,22)(19,26)(20,30)(24,34)(25,36)(27,35)(29,37)(31,40)(32,39)(33,38)(41,42)(43,44);(2,4)(13,15)(14,19)(16,29)(17,26)(18,25)(20,37)(21,23)(22,35)(24,31)(27,32)(28,44)(30,43)(33,42)(34,38)(36,39)(40,41);(5,7,9,8)(6,10,11,12)(13,16,17,30)(14,20,21,28)(15,24,26,42)(18,33,32,31)(19,34,23,41)(22,40,39,38)(25,29,27,43)(35,37,36,44);(5,8,9,7)(6,10,11,12)(15,25)(18,32)(19,35)(22,39)(23,36)(24,29)(26,27)(31,33)(34,37)(38,40)(41,44)(42,43);(14,21)(15,26)(20,28)(22,39)(24,42)(25,27)(29,43)(38,40)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359132);