Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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)(5,6)(7,10)(8,12)(9,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30)(31,37)(32,39)(33,36)(34,35)(38,44)(40,42)(41,43);(5,7,9,12)(6,8,11,10)(16,24)(19,26)(22,28)(25,27)(30,35)(32,37)(33,41)(38,42);(7,12)(8,10)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,31)(30,32)(33,42)(34,39)(35,37)(36,44)(38,41)(40,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,44)(42,43);(5,8,9,10)(6,7,11,12)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)(29,33,34,41)(30,36,35,43)(31,38,39,42)(32,40,37,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgh,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989838);