Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2003 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)(7,9)(8,10)(11,12)(13,14,19,23,20,24,18,17)(15,25,27,35,28,38,33,21)(16,22,30,36,31,37,32,29)(26,39,41,44,42,43,40,34);(5,6)(7,11)(8,10)(9,12)(13,15,20,28)(14,21,24,35)(16,26,31,42)(17,25,23,38)(18,27,19,33)(22,34,37,44)(29,39,36,43)(30,40,32,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,29)(18,32)(19,30)(20,31)(21,34)(23,36)(24,37)(25,39)(27,41)(28,42)(33,40)(35,44)(38,43);(7,12)(9,11)(13,17,18,24,20,23,19,14)(15,21,33,38,28,35,27,25)(16,29,32,37,31,36,30,22)(26,34,40,43,42,44,41,39);(6,10)(9,11)(13,18,20,19)(14,17,24,23)(15,27,28,33)(16,30,31,32)(21,25,35,38)(22,36,37,29)(26,40,42,41)(34,43,44,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913950);