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 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,17,18)(15,21,23,19)(16,20,25,26)(22,28,24,27)(29,30)(31,32)(33,41)(34,35)(36,39)(37,43)(38,42)(40,44);(2,4)(5,6,9,8)(7,10,11,12)(14,18)(19,21)(20,26)(27,28)(30,35)(32,39)(37,41)(40,42);(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,31)(30,32)(33,38)(34,36)(35,39)(37,40)(41,42)(43,44);(5,7)(6,10)(8,12)(9,11)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27)(29,32)(30,36)(31,35)(33,42)(34,39)(37,38)(40,43)(41,44);(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,33,34,43)(30,37,35,41)(31,38,36,44)(32,40,39,42)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=gh,d^2=g,e^2=i,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, 2698571);