Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,18,19)(15,20,22,16)(17,21,25,27)(23,28,26,24)(29,30,34,38)(31,32,36,35)(33,37,42,44)(39,40,41,43);(6,9)(10,12)(14,19)(15,22)(17,25)(24,28)(29,31,34,36)(30,35,38,32)(33,39,42,41)(37,43,44,40);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,26)(18,22)(21,24)(23,25)(27,28)(29,31)(30,35)(32,38)(33,41)(34,36)(37,40)(39,42)(43,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,25,28)(21,23,27,26)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,41,44,39);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,21,19,27)(15,23,22,26)(16,24,20,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,41)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382408);