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 | |
1626 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)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,20,21,16)(17,22,25,26)(23,27,28,24)(29,30,35,33)(31,37,32,36)(34,41,42,38)(39,43,40,44);(7,11)(10,12)(14,19)(15,21)(17,25)(24,27)(29,31,35,32)(30,36,33,37)(34,40,42,39)(38,44,41,43);(5,7)(6,10)(8,12)(9,11)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,26)(25,28)(29,32,35,31)(30,37,33,36)(34,40,42,39)(38,44,41,43);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23)(29,33,35,30)(31,36,32,37)(34,41,42,38)(39,43,40,44);(13,17,18,25)(14,22,19,26)(15,23,21,28)(16,24,20,27)(29,34,35,42)(30,38,33,41)(31,39,32,40)(36,43,37,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=f,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, 1611905);