Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,28,38)(16,21,29,35)(17,23,18,22)(20,32,34,27)(26,39,40,44)(30,37,31,36)(33,42,43,41);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,25)(23,38)(24,34)(29,40)(30,41)(31,42)(35,43)(36,39)(37,44);(2,4)(6,10)(9,12)(13,16)(14,21)(15,26)(17,30)(18,31)(19,29)(20,33)(22,36)(23,37)(24,35)(25,39)(27,41)(28,40)(32,42)(34,43)(38,44);(2,4)(16,29)(21,35)(26,40)(30,31)(33,43)(36,37)(39,44)(41,42);(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,30,29,31)(20,25,34,38)(21,36,35,37)(26,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347518);