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 | |
1608 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)(13,14,18,19)(15,16,23,20)(17,25,26,21)(22,27,28,24)(29,30)(31,35)(32,39)(33,34)(36,41)(37,42)(38,44)(40,43);(5,6,9,8)(7,11,12,10)(14,19)(16,20)(17,26)(22,28)(30,34)(31,37)(36,39)(38,40);(5,7,9,12)(6,10,8,11)(13,15,18,23)(14,20,19,16)(17,22,26,28)(21,27,25,24)(29,31,33,37)(30,35,34,42)(32,40,41,38)(36,44,39,43);(1,2,3,4)(13,16,18,20)(14,15,19,23)(17,24,26,27)(21,22,25,28)(29,32)(30,36)(31,38)(33,41)(34,39)(35,43)(37,40)(42,44);(5,8,9,6)(7,11,12,10)(13,17,18,26)(14,21,19,25)(15,22,23,28)(16,24,20,27)(29,30)(31,35)(32,36)(33,34)(37,42)(38,43)(39,41)(40,44)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=gh,d^2=fgi,e^2=fh,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, 1489878);