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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1857 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)(9,12)(13,14)(15,21)(16,18)(17,24)(19,22)(20,25)(23,27)(26,28)(29,30,34,35)(31,36,37,32)(33,38,42,43)(39,44,41,40);(5,6,8,10)(7,9,11,12)(14,16)(15,22)(17,25)(27,28)(30,35)(32,36)(38,43)(40,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,21)(17,26)(18,22)(20,28)(23,25)(24,27)(29,31)(30,36)(32,35)(33,41)(34,37)(38,40)(39,42)(43,44);(1,2,3,4)(13,16,18,14)(15,21,22,19)(17,24,25,20)(23,28,26,27)(29,32,34,36)(30,37,35,31)(33,40,42,44)(38,41,43,39);(5,7,8,11)(6,9,10,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,27)(21,28)(22,26)(29,33)(30,38)(31,39)(32,40)(34,42)(35,43)(36,44)(37,41)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=i,e^2=gh,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, 4990572);