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 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,10,8)(7,11,9,12)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28)(29,30,34,35)(31,36,37,32)(33,38,41,42)(39,43,44,40);(2,4)(5,7,10,9)(6,11,8,12)(14,15)(16,23)(17,25)(26,27)(30,35)(31,37)(33,41)(40,43);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,42)(41,44);(5,9,10,7)(6,11,8,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,32,34,36)(30,37,35,31)(33,40,41,43)(38,44,42,39);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,33,34,41)(30,38,35,42)(31,39,37,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fgh,d^2=gi,e^2=fg,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, 1724060);