Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1626 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)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,20,23,27)(16,24,26,17)(21,22,25,28)(30,34)(31,37)(32,40)(42,44);(7,11)(10,12)(14,19)(17,24)(20,27)(22,28)(29,30,33,34)(31,35,37,38)(32,41,40,36)(39,44,43,42);(5,7,9,11)(6,10,8,12)(13,15)(14,20)(16,25)(17,22)(18,23)(19,27)(21,26)(24,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,2,3,4)(13,16)(14,17)(15,21)(18,26)(19,24)(20,28)(22,27)(23,25)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(5,8,9,6)(7,12,11,10)(13,17)(14,16)(15,22)(18,24)(19,26)(20,25)(21,27)(23,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1654125);