Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1598 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)(13,14,18,17)(15,21,23,19)(16,24,25,20)(22,26,28,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(6,9)(10,12)(16,25)(20,24)(22,28)(26,27)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,15,18,23)(14,19,17,21)(16,22,25,28)(20,26,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2,3,4)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,14)(15,21,23,19)(16,24,25,20)(22,27,28,26)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fgh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4407626);