Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1994 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)(7,11)(10,12)(13,14,19,24)(15,25,28,38)(16,21,30,36)(17,23,18,22)(20,32,34,27)(26,39,42,43)(29,37,31,35)(33,40,44,41);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,34)(16,26,30,42)(17,27,18,32)(21,33,36,44)(22,25,23,38)(29,40,31,41)(35,43,37,39);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,29)(19,30)(20,33)(22,37)(23,35)(24,36)(25,39)(27,41)(28,42)(32,40)(34,44)(38,43);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,35,36,37)(26,40,42,41)(33,43,44,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050172);