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 | |
1640 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)(13,14,18,17)(15,19,23,22)(16,24,26,20)(21,28,25,27)(29,30)(31,39)(32,42)(33,41)(34,35)(36,40)(37,43)(38,44);(5,7,9,12)(6,10,8,11)(14,17)(19,22)(20,24)(27,28)(30,35)(31,40)(32,43)(33,44);(5,7)(6,10)(8,11)(9,12)(13,15,18,23)(14,19,17,22)(16,25,26,21)(20,28,24,27)(29,31)(30,36)(32,38)(33,42)(34,40)(35,39)(37,44)(41,43);(1,2,3,4)(13,16)(14,20)(15,21)(17,24)(18,26)(19,27)(22,28)(23,25)(29,32)(30,37)(31,41)(33,36)(34,43)(35,42)(38,40)(39,44);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,22,23,19)(16,24,26,20)(21,28,25,27)(29,33)(30,38)(31,42)(32,36)(34,44)(35,41)(37,40)(39,43)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,d^2=fi,e^2=fh,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, 1587991);