Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1696 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)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,20,23,27)(16,24,25,17)(21,22,28,26)(30,31)(32,39)(33,41)(43,44);(7,11)(10,12)(14,19)(16,25)(20,27)(21,28)(29,30,34,31)(32,35,39,37)(33,36,41,38)(40,43,42,44);(5,7,9,11)(6,10,8,12)(13,15)(14,20)(16,21)(17,26)(18,23)(19,27)(22,24)(25,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,43,42,44);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,16,18,25)(14,17,19,24)(15,21,23,28)(20,26,27,22)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(5,8,9,6)(7,12,11,10)(13,17)(14,16)(15,22)(18,24)(19,25)(20,28)(21,27)(23,26)(29,33)(30,36)(31,38)(32,40)(34,41)(35,43)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=gh,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3386954);