Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1610 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)(13,14)(15,21)(16,23)(17,25)(18,19)(20,26)(22,24)(27,28)(29,30,35,34)(31,38,32,36)(33,41,42,37)(39,43,40,44);(6,9)(10,12)(14,19)(16,21)(17,26)(22,28)(29,31,35,32)(30,36,34,38)(33,39,42,40)(37,43,41,44);(5,6,8,9)(7,10,11,12)(13,15,18,23)(14,16,19,21)(17,22,26,28)(20,24,25,27)(29,32,35,31)(30,36,34,38)(33,39,42,40)(37,44,41,43);(1,2,3,4)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,33,35,42)(30,37,34,41)(31,39,32,40)(36,43,38,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,25)(21,27)(23,28)(29,34,35,30)(31,36,32,38)(33,41,42,37)(39,43,40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fh,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1522612);