Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2323 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,32)(26,34)(27,36)(29,35)(31,38)(33,37)(39,40)(41,43)(42,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,27,19,29,17,28)(14,20,22,35,24,36,23,25)(16,26,32,40,33,42,31,41)(21,34,30,43,38,44,37,39);(7,12)(9,10)(13,16,17,31,19,33,18,32)(14,21,23,37,24,38,22,30)(15,26,28,41,29,42,27,40)(20,34,25,39,36,44,35,43);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,40,42,41)(34,39,44,43);(7,12)(9,10)(13,17,19,18)(14,22,24,23)(15,28,29,27)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,41,42,40)(34,43,44,39)Polycyclic presentation:
2^9|b^2=fhi,c^2=gi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514189);