Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,19,22)(15,20,26,36)(16,29,31,38)(17,24,18,23)(21,32,37,30)(25,39,40,44)(27,35,28,34)(33,42,43,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,36)(23,34)(24,35)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(6,9)(7,11)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,39)(35,44)(36,43);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,37)(23,24)(25,40)(27,28)(29,38)(41,42);(6,9)(10,12)(13,17,19,18)(14,23,22,24)(15,27,26,28)(16,30,31,32)(20,34,36,35)(21,29,37,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420675);