Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3122 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)(7,11)(10,12)(13,14,19,25)(15,22,29,36)(16,30,34,40)(17,24,18,23)(20,27,37,28)(21,33,41,32)(26,38,31,35)(39,42,44,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,25,37)(16,31,34,26)(17,27,18,28)(21,38,41,35)(22,23,36,24)(30,39,40,44)(32,43,33,42);(6,9)(10,12)(13,16,19,34)(14,21,25,41)(15,26,29,31)(17,32,18,33)(20,35,37,38)(22,39,36,44)(23,30,24,40)(27,42,28,43);(6,9)(10,12)(13,15)(14,22)(16,26)(17,28)(18,27)(19,29)(20,24)(21,39)(23,37)(25,36)(30,38)(31,34)(32,43)(33,42)(35,40)(41,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,32,34,33)(20,36,37,22)(21,30,41,40)(26,42,31,43)(35,44,38,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398338);