Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #45388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,26,29,21)(16,30,34,41)(17,36,28,38)(18,25,20,23)(22,35,40,33)(27,39,43,44)(31,37,32,42);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,31)(17,20)(18,28)(19,29)(22,39)(23,38)(24,26)(25,36)(27,35)(30,42)(32,34)(33,43)(37,41)(40,44);(6,9)(10,11)(13,16,20,35,19,34,18,33)(14,22,25,41,24,40,23,30)(15,27,28,31,29,43,17,32)(21,37,38,39,26,42,36,44);(5,7,8,12)(6,10,9,11)(13,17,18,29,19,28,20,15)(14,21,23,36,24,26,25,38)(16,32,33,43,34,31,35,27)(22,37,30,44,40,42,41,39);(1,3,4,2)(13,18,19,20)(14,23,24,25)(15,28,29,17)(16,33,34,35)(21,38,26,36)(22,30,40,41)(27,31,43,32)(37,39,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504008);