Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
1866 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)(14,20)(15,25)(17,18)(21,37)(22,36)(23,24)(26,35)(27,30)(29,40)(31,32)(38,44)(39,41);(5,6)(7,11)(8,10)(9,12)(13,14,18,20,19,24,17,23)(15,26,28,37,25,21,34,35)(16,29,32,40,33,36,31,22)(27,41,42,38,30,44,43,39);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,34)(18,28)(19,25)(20,35)(22,39)(23,37)(24,26)(27,33)(29,41)(31,42)(32,43)(36,44)(38,40);(7,12)(9,11)(13,16,19,33)(14,22,24,40)(15,27,25,30)(17,32,18,31)(20,36,23,29)(21,38,26,39)(28,43,34,42)(35,41,37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,28,25,34)(16,31,33,32)(21,35,26,37)(22,36,40,29)(27,42,30,43)(38,44,39,41)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910534);