Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,32,35)(15,25,27,22,28,39,17,20)(16,21,31,37,30,36,42,43)(26,40,29,38,41,44,33,34);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,24,22)(16,26,30,41)(17,32,27,18)(21,34,36,38)(23,39,35,25)(29,31,33,42)(37,44,43,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,33)(18,31)(19,30)(20,34)(22,38)(23,37)(24,36)(25,40)(27,29)(28,41)(32,42)(35,43)(39,44);(7,12)(9,11)(13,17,18,15,19,27,32,28)(14,22,23,39,24,20,35,25)(16,29,31,41,30,33,42,26)(21,34,37,40,36,38,43,44);(6,10)(9,11)(16,30)(21,36)(26,41)(29,33)(31,42)(34,38)(37,43)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903923);