Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1670 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)(6,7)(8,12)(13,14,18,21)(15,22,16,19)(17,25,26,20)(23,27,24,28)(30,34)(31,37)(32,39)(43,44);(6,7)(10,11)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28)(29,30,33,34)(31,35,37,41)(32,36,39,42)(38,43,40,44);(5,6,9,7)(8,10,12,11)(13,16,18,15)(14,19,21,22)(17,24,26,23)(20,27,25,28)(29,31)(30,35)(32,40)(33,37)(34,41)(36,44)(38,39)(42,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,43)(39,42)(40,44);(5,8,9,12)(6,10,7,11)(13,14)(15,19)(16,22)(17,20)(18,21)(23,27)(24,28)(25,26)(29,32)(30,36)(31,38)(33,39)(34,42)(35,43)(37,40)(41,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fg,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4576971);