Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,32,18,22,17,23)(15,24,31,20,26,39,27,35)(16,21,33,44,29,37,30,38)(25,40,43,34,28,42,41,36);(7,8)(10,12)(13,15)(14,20)(16,28)(17,31)(18,26)(19,27)(21,36)(22,35)(23,39)(24,32)(25,29)(30,41)(33,43)(34,37)(38,40)(42,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,33)(20,34)(22,37)(23,38)(24,40)(26,28)(27,41)(31,43)(32,44)(35,36)(39,42);(5,8,9,7)(6,10,11,12)(14,22)(15,26)(21,37)(23,32)(25,28)(27,31)(38,44)(41,43);(16,29)(21,37)(25,28)(30,33)(34,36)(38,44)(40,42)(41,43)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370202);