Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 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)(7,11)(8,12)(9,10)(13,14,19,24)(15,25,16,21)(17,22,32,40)(18,33,20,23)(26,42,28,35)(27,36,29,37)(30,44,43,38)(31,41,34,39);(1,2)(3,4)(6,10)(11,12)(13,15,19,16)(14,21,24,25)(17,30,32,43)(18,27,20,29)(22,38,40,44)(23,36,33,37)(26,34,28,31)(35,39,42,41);(2,4)(13,16,19,15)(14,21,24,25)(17,28,32,26)(18,27,20,29)(22,35,40,42)(23,37,33,36)(30,31,43,34)(38,39,44,41);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,22)(15,26)(16,28)(18,31)(19,32)(20,34)(21,35)(23,39)(24,40)(25,42)(27,43)(29,30)(33,41)(36,38)(37,44);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,23,24,33)(15,27,16,29)(17,31,32,34)(21,36,25,37)(22,39,40,41)(26,43,28,30)(35,38,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7428361);