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, 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:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,25,27)(16,21,22,17)(23,24,28,26)(29,30,34,37)(31,35,39,32)(33,36,42,44)(38,43,41,40);(7,11)(10,12)(14,19)(17,21)(20,27)(24,26)(29,31,34,39)(30,32,37,35)(33,41,42,38)(36,43,44,40);(6,9)(10,12)(13,15,18,25)(14,20,19,27)(16,23,22,28)(17,26,21,24)(29,31)(30,35)(32,37)(33,41)(34,39)(36,40)(38,42)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,22)(14,21,19,17)(15,23,25,28)(20,24,27,26)(29,32,34,35)(30,31,37,39)(33,40,42,43)(36,38,44,41);(5,7,8,11)(6,10,9,12)(13,17,18,21)(14,22,19,16)(15,24,25,26)(20,23,27,28)(29,33,34,42)(30,36,37,44)(31,38,39,41)(32,40,35,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,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, 5382872);