Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,29)(17,22)(18,23)(19,32)(20,26)(21,30)(25,40)(27,39)(28,35)(31,36)(33,37)(34,43)(38,42)(41,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,34)(22,35)(23,24)(29,44)(30,42)(31,41)(32,39)(33,43)(36,40)(37,38);(2,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,33)(20,34)(22,36)(23,37)(24,38)(26,41)(27,42)(28,43)(29,32)(35,40)(39,44);(1,2)(3,4)(5,7,9,8)(6,11,12,10)(15,26)(20,24)(25,41)(27,28)(34,38)(35,39)(40,44)(42,43);(2,4)(5,8,9,7)(6,10,12,11)(13,17,18,19)(14,22,23,32)(15,27,26,28)(16,30,31,33)(20,35,24,39)(21,36,37,29)(25,42,41,43)(34,40,38,44)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485984);