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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1806 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)(13,14,18,23,19,24,17,31)(15,20,27,37,28,38,32,36)(16,22,29,41,30,40,33,21)(25,42,44,39,26,35,43,34);(1,2)(3,4)(6,9)(7,11)(13,15,19,28)(14,20,24,38)(16,26,30,25)(17,32,18,27)(21,39,41,34)(22,35,40,42)(23,37,31,36)(29,43,33,44);(5,6)(7,10)(8,9)(11,12)(13,16,19,30)(14,21,24,41)(15,25,28,26)(17,29,18,33)(20,34,38,39)(22,23,40,31)(27,43,32,44)(35,36,42,37);(2,4)(6,9)(10,12)(13,16)(14,22)(15,26)(17,33)(18,29)(19,30)(20,35)(21,31)(23,41)(24,40)(25,28)(27,43)(32,44)(34,37)(36,39)(38,42);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,31)(15,27,28,32)(16,29,30,33)(20,36,38,37)(21,40,41,22)(25,43,26,44)(34,42,39,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7055391);