Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #24428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2143 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,24)(18,23)(19,25)(20,28)(21,29)(27,39)(30,36)(31,38)(32,42)(33,41)(34,37)(35,40)(43,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,25,37)(16,30,33,43)(17,28,18,34)(21,23,26,24)(22,39,41,38)(27,32,31,35)(36,40,44,42);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,31)(17,34)(18,28)(19,29)(20,24)(22,39)(23,37)(25,26)(27,33)(30,32)(35,43)(36,40)(38,41)(42,44);(7,12)(9,11)(13,16)(14,22)(15,27)(17,32)(18,35)(19,33)(20,36)(21,38)(23,40)(24,42)(25,41)(26,39)(28,30)(29,31)(34,43)(37,44);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,28,29,34)(16,32,33,35)(20,26,37,21)(22,40,41,42)(27,30,31,43)(36,39,44,38)Polycyclic presentation:
2^9|b^2=i,c^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961956);