Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3413 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)(7,10)(8,9)(11,12)(13,14,20,26)(15,27,16,22)(17,23,35,42)(18,24,19,25)(21,30,37,29)(28,39,31,38)(32,36,43,44)(33,40,34,41);(1,2,3,4)(13,15,18,29,20,16,19,30)(14,21,25,27,26,37,24,22)(17,32,34,28,35,43,33,31)(23,39,40,36,42,38,41,44);(6,9)(10,12)(13,16,20,15)(14,22,26,27)(17,31,35,28)(18,29,19,30)(21,25,37,24)(23,38,42,39)(32,34,43,33)(36,41,44,40);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,28)(16,31)(18,33)(19,34)(20,35)(21,36)(22,38)(24,40)(25,41)(26,42)(27,39)(29,32)(30,43)(37,44);(13,18,20,19)(14,24,26,25)(15,29,16,30)(17,33,35,34)(21,22,37,27)(23,40,42,41)(28,32,31,43)(36,38,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356451);