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 | |
1642 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)(15,18)(16,21)(17,22)(19,23)(20,25)(24,26)(27,28)(29,30,35,32)(31,37,33,36)(34,38,42,40)(39,44,41,43);(6,9)(10,12)(14,15)(16,23)(17,25)(26,27)(29,31,35,33)(30,36,32,37)(34,41,42,39)(38,44,40,43);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,32,35,30)(31,36,33,37)(34,40,42,38)(39,43,41,44);(1,2,3,4)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,33,35,31)(30,37,32,36)(34,41,42,39)(38,44,40,43);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,34,35,42)(30,38,32,40)(31,39,33,41)(36,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=gi,e^2=gh,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, 1522602);