Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,26)(22,27)(25,28)(29,30,34,36)(31,37,39,32)(33,41,43,35)(38,40,44,42);(6,9)(10,12)(14,18)(19,21)(20,24)(27,28)(29,31,34,39)(30,32,36,37)(33,38,43,44)(35,40,41,42);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,31)(30,32)(33,38)(34,39)(35,40)(36,37)(41,42)(43,44);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,32)(30,31)(33,42)(34,37)(35,44)(36,39)(38,41)(40,43);(5,7,8,11)(6,10,9,12)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33,34,43)(30,35,36,41)(31,38,39,44)(32,40,37,42)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773472);