Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,20,25)(15,26,29,22)(16,21,28,37)(17,23,33,42)(18,24,19,35)(27,39,43,38)(30,36,31,44)(32,40,34,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,28,20,29,19,16)(14,21,35,22,25,37,24,26)(17,27,34,31,33,43,32,30)(23,36,40,38,42,44,41,39);(6,9)(7,11)(13,16,19,29,20,28,18,15)(14,22,24,21,25,26,35,37)(17,31,32,27,33,30,34,43)(23,39,41,44,42,38,40,36);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,30)(18,34)(19,32)(20,33)(21,36)(22,38)(24,41)(25,42)(26,39)(28,31)(29,43)(35,40)(37,44);(6,9)(10,12)(13,18,20,19)(14,24,25,35)(15,28,29,16)(17,32,33,34)(21,26,37,22)(23,40,42,41)(27,30,43,31)(36,38,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6427758);