Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1594 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)(7,11)(10,12)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,28)(26,27)(29,30,34,38)(31,33,37,35)(32,40,43,36)(39,44,41,42);(6,9)(10,12)(14,19)(15,20)(21,27)(22,25)(29,31,34,37)(30,35,38,33)(32,41,43,39)(36,42,40,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,24)(17,25,26,22)(21,28,27,23)(29,31)(30,35)(32,39)(33,38)(34,37)(36,44)(40,42)(41,43);(1,2,3,4)(13,16)(14,20)(15,19)(17,23)(18,24)(21,25)(22,27)(26,28)(29,32)(30,36)(31,39)(33,42)(34,43)(35,44)(37,41)(38,40);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,21,19,27)(15,22,20,25)(16,23,24,28)(29,33,34,35)(30,37,38,31)(32,42,43,44)(36,41,40,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4571754);