Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1634 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)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,35)(31,32,37,40)(33,41,43,36)(38,42,44,39);(5,7,10,9)(6,11,8,12)(15,21)(18,25)(22,27)(26,28)(30,35)(31,37)(36,41)(38,44);(5,6,10,8)(7,12,9,11)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28)(29,31)(30,32)(33,38)(34,37)(35,40)(36,39)(41,42)(43,44);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,42)(34,40)(35,37)(36,44)(38,41)(39,43);(5,9,10,7)(6,11,8,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,43)(30,36,35,41)(31,38,37,44)(32,39,40,42)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773592);