Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3554 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,24)(16,28)(17,23)(18,22)(19,33)(20,26)(21,31)(25,41)(27,37)(29,44)(30,39)(32,36)(34,40)(35,42)(38,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,32)(18,26)(19,27)(21,38)(22,24)(23,37)(25,30)(28,41)(31,42)(33,36)(34,43)(35,40)(39,44);(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,30)(19,34)(20,35)(22,40)(23,39)(24,38)(26,29)(27,42)(28,33)(32,43)(36,44)(37,41);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(16,30)(23,33)(27,32)(31,34)(35,38)(41,44);(6,9)(10,12)(13,17,18,19)(14,23,22,33)(15,27,26,32)(16,31,30,34)(20,36,24,37)(21,39,40,28)(25,42,29,43)(35,44,38,41)Polycyclic presentation:
2^9|a^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420132);