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 | |
1670 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)(15,22)(16,20)(17,25)(18,19)(21,26)(23,27)(24,28)(29,30,34,36)(31,37,40,35)(32,41,43,33)(38,44,42,39);(6,9)(10,12)(13,15,19,16)(14,20,18,22)(17,23,26,24)(21,27,25,28)(29,31,34,40)(30,35,36,37)(32,42,43,38)(33,39,41,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,15)(14,20,18,22)(17,24,26,23)(21,27,25,28)(29,31)(30,35)(32,38)(33,44)(34,40)(36,37)(39,41)(42,43);(1,2,3,4)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)(29,32)(30,33)(31,38)(34,43)(35,44)(36,41)(37,39)(40,42);(5,7,8,11)(6,10,9,12)(13,18,19,14)(15,22,16,20)(17,25,26,21)(23,28,24,27)(29,33)(30,32)(31,39)(34,41)(35,42)(36,43)(37,38)(40,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fi,e^2=fh,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, 4575962);