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 | |
1684 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)(5,6,10,8)(7,11,9,12)(13,14)(15,19)(16,23)(17,18)(20,25)(21,28)(22,26)(24,27)(29,30,34,35)(31,39,40,36)(32,42,43,37)(33,41,44,38);(5,7,10,9)(6,11,8,12)(14,18)(19,26)(20,23)(27,28)(30,35)(31,40)(32,43)(33,44);(5,8,10,6)(7,11,9,12)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27)(29,31,34,40)(30,36,35,39)(32,38,43,41)(33,42,44,37);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,16)(14,20)(15,21)(17,25)(18,23)(19,27)(22,24)(26,28)(29,32,34,43)(30,37,35,42)(31,41,40,38)(33,36,44,39);(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,25)(24,28)(29,33,34,44)(30,38,35,41)(31,42,40,37)(32,36,43,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=fgi,e^2=g,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, 1554243);