Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,24,26,19)(16,21,28,36)(18,20,30,23)(25,37,38,32)(27,33,40,31)(29,34,41,35)(39,43,44,42);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,31)(21,32)(22,24)(23,33)(28,38)(29,39)(30,40)(34,42)(35,43)(36,37)(41,44);(2,4)(14,20)(16,28)(18,30)(19,31)(21,35)(22,23)(24,33)(25,38)(27,40)(32,43)(34,36)(37,42);(2,4)(13,16)(14,21)(15,25)(17,28)(18,29)(19,32)(20,34)(22,36)(23,35)(24,37)(26,38)(27,39)(30,41)(31,42)(33,43)(40,44);(5,7,8,11)(6,10,9,12)(15,26)(16,28)(19,24)(21,36)(27,40)(29,41)(31,33)(34,35)Polycyclic presentation:
2^9|a^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347413);