Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14)(15,24)(16,28)(17,23)(18,22)(19,32)(20,26)(21,29)(25,40)(27,36)(30,38)(31,35)(33,37)(34,42)(39,43)(41,44);(1,2)(3,4)(6,10)(11,12)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,34)(22,24)(23,36)(28,44)(29,43)(30,41)(32,35)(33,42)(37,39)(38,40);(2,4)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,34)(22,37)(23,38)(24,39)(26,41)(27,42)(28,32)(31,43)(35,44)(36,40);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(14,22)(15,26)(21,37)(23,32)(25,41)(27,31)(28,38)(42,43);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,19)(14,23,22,32)(15,27,26,31)(16,29,30,33)(20,35,24,36)(21,38,37,28)(25,42,41,43)(34,44,39,40)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485457);