Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,21)(17,20)(19,24)(22,26)(23,28)(25,27)(29,30,34,31)(32,37,39,35)(33,38,42,36)(40,43,41,44);(2,4)(5,7,9,6)(8,11,12,10)(14,15)(19,21)(20,22)(27,28)(30,31)(32,39)(36,38)(40,41);(1,2)(3,4)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,31,34,30)(32,35,39,37)(33,38,42,36)(40,43,41,44);(5,8,9,12)(6,10,7,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,39)(30,35,31,37)(33,41,42,40)(36,44,38,43);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2686687);