Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,39)(26,40)(27,38)(28,35)(29,36)(30,37)(31,34)(32,41)(33,42)(43,44);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,25)(17,24)(18,31)(20,23)(21,32)(22,38)(26,30)(27,29)(28,41)(33,37)(34,36)(35,39)(40,44)(42,43);(2,4)(5,8,9,12)(6,10,7,11)(15,24)(17,30)(18,29)(19,32)(21,37)(22,36)(23,39)(25,41)(26,27)(28,43)(31,42)(33,34)(35,44)(38,40);(2,4)(6,7)(10,11)(14,20)(15,25)(21,35)(22,36)(24,41)(26,42)(27,31)(37,44)Polycyclic presentation:
2^9|b^2=ei,c^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416630);