Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 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)(7,11)(10,12)(13,14,19,24)(15,25,30,43)(16,21,31,39)(17,22,33,40)(18,34,26,23)(20,35,38,29)(27,37,44,41)(28,42,32,36);(5,6)(7,10)(8,9)(11,12)(13,15,26,35,19,30,18,29)(14,20,34,43,24,38,23,25)(16,27,33,28,31,44,17,32)(21,36,22,41,39,42,40,37);(2,4)(6,9)(10,12)(13,16)(14,21)(15,27)(17,18)(19,31)(20,36)(22,34)(23,40)(24,39)(25,37)(26,33)(28,35)(29,32)(30,44)(38,42)(41,43);(6,9)(10,12)(13,17)(14,22)(15,28)(16,26)(18,31)(19,33)(20,37)(21,23)(24,40)(25,42)(27,29)(30,32)(34,39)(35,44)(36,43)(38,41);(5,7,8,11)(6,10,9,12)(16,31)(17,33)(21,39)(22,40)(27,44)(28,32)(36,42)(37,41)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398995);