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 | |
1612 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)(15,21)(16,24)(17,18)(19,22)(20,25)(23,27)(26,28)(29,30,34,35)(31,36,40,42)(32,39,43,37)(33,38,44,41);(5,6,9,8)(7,11,12,10)(14,17)(15,22)(16,25)(27,28)(30,35)(33,44)(36,42)(37,39);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,26)(17,21)(18,22)(20,28)(23,25)(24,27)(29,31)(30,36)(32,41)(33,39)(34,40)(35,42)(37,44)(38,43);(1,2,3,4)(13,16)(14,20)(15,23)(17,24)(18,25)(19,27)(21,28)(22,26)(29,32,34,43)(30,37,35,39)(31,38,40,41)(33,42,44,36);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,21,22,19)(16,24,25,20)(23,28,26,27)(29,33)(30,38)(31,39)(32,42)(34,44)(35,41)(36,43)(37,40)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=gi,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, 1588553);