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 | |
1628 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)(7,10,12,11)(13,14,18,21)(15,16,22,19)(17,25,26,20)(23,27,28,24)(29,30,34,33)(31,35,37,39)(32,40,42,36)(38,44,41,43);(5,6,9,8)(7,11,12,10)(15,22)(16,19)(23,28)(24,27)(30,33)(31,37)(36,40)(38,41);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,21)(17,23)(18,22)(20,27)(24,25)(26,28)(29,31)(30,35)(32,41)(33,39)(34,37)(36,44)(38,42)(40,43);(1,2,3,4)(13,16,18,19)(14,15,21,22)(17,24,26,27)(20,23,25,28)(29,32)(30,36)(31,38)(33,40)(34,42)(35,43)(37,41)(39,44);(5,8,9,6)(7,11,12,10)(13,17,18,26)(14,20,21,25)(15,23,22,28)(16,24,19,27)(29,33,34,30)(31,39,37,35)(32,40,42,36)(38,44,41,43)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=fi,e^2=h,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, 1555600);