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)(16,20)(17,21)(18,22)(19,26)(24,41)(25,35)(27,34)(28,44)(29,37)(30,38)(31,39)(32,40)(33,42)(36,43);(2,4)(13,15)(14,19)(16,28)(17,26)(18,25)(20,36)(21,23)(22,34)(24,30)(27,31)(29,43)(32,42)(33,38)(35,39)(37,44)(40,41);(5,7,9,8)(6,10,11,12)(13,16,17,29)(14,20,21,37)(15,24,26,42)(18,32,31,30)(19,33,23,41)(22,40,39,38)(25,28,27,43)(34,36,35,44);(5,8,9,7)(6,10,11,12)(15,25)(18,31)(19,34)(22,39)(23,35)(24,28)(26,27)(30,32)(33,36)(38,40)(41,44)(42,43);(14,21)(15,26)(20,37)(22,39)(24,42)(25,27)(28,43)(38,40)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=g,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, 6359130);