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 | |
1642 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)(15,20)(16,17)(18,19)(21,26)(22,28)(23,25)(24,27)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(7,11)(10,12)(14,19)(17,21)(20,27)(23,28)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,7)(6,10)(8,12)(9,11)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,28)(22,26)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2,3,4)(13,16)(14,21)(15,22)(17,19)(18,26)(20,23)(24,25)(27,28)(29,30)(31,36)(32,35)(33,37)(34,38)(39,44)(40,43)(41,42);(5,8,9,6)(7,12,11,10)(13,17,18,21)(14,16,19,26)(15,23,24,28)(20,25,27,22)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,44)(38,41)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=fi,e^2=fgh,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, 1620757);