Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2386 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,21,31,37)(15,23,26,32,24,38,30,19)(16,20,29,35,28,36,43,44)(25,39,40,34,27,41,42,33);(5,7)(6,9)(8,11)(10,12)(13,15,18,24)(14,19,21,32)(16,27,28,25)(17,30,31,26)(20,34,36,33)(22,38,37,23)(29,40,43,42)(35,39,44,41);(1,2,3,4)(6,10)(7,11)(15,24)(19,32)(23,38)(25,27)(26,30)(33,34)(39,41)(40,42);(7,11)(9,12)(13,16)(14,20)(15,25)(17,29)(18,28)(19,33)(21,36)(22,35)(23,39)(24,27)(26,40)(30,42)(31,43)(32,34)(37,44)(38,41);(14,21)(16,28)(19,32)(22,37)(23,38)(25,27)(29,43)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366781);