Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 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)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,34)(26,38)(28,39)(29,35)(30,36)(31,37)(33,41)(40,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,15,16,25,19,28,18,27)(14,20,32,39,23,34,21,24)(17,26,29,40,31,42,30,41)(22,33,36,43,37,44,35,38);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,18)(14,21,23,32)(15,25,28,27)(17,30,31,29)(20,24,34,39)(22,36,37,35)(26,41,42,40)(33,43,44,38);(7,12)(9,11)(13,17,19,31)(14,22,23,37)(15,26,28,42)(16,29,18,30)(20,33,34,44)(21,35,32,36)(24,38,39,43)(25,40,27,41);(6,10)(9,11)(17,31)(22,37)(26,42)(29,30)(33,44)(35,36)(38,43)(40,41)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410189);