Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,31)(26,40)(28,36)(29,35)(32,38)(33,37)(34,43)(39,41)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,28)(18,27)(19,29)(21,34)(22,25)(23,35)(24,36)(30,44)(31,42)(32,41)(33,43)(37,39)(38,40);(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,30)(24,38)(25,39)(27,41)(28,42)(29,43)(35,44)(36,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,35,36,25)(21,37,38,30)(26,41,43,42)(34,44,40,39);(7,11)(10,12)(13,17,19,18)(14,22,24,23)(15,28,29,27)(16,31,33,32)(20,25,36,35)(21,37,38,30)(26,42,43,41)(34,39,40,44)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6440754);