Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,23,35,41)(15,25,28,38,27,43,31,20)(16,29,33,37,32,21,39,40)(17,34,26,44,36,22,30,42);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,38)(16,30,32,26)(17,33,36,39)(18,31,35,28)(21,34,29,22)(24,43,41,25)(37,44,40,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,21,23,29)(15,26,27,30)(17,28,36,31)(18,33,35,39)(20,22,38,34)(24,40,41,37)(25,42,43,44);(7,12)(9,11)(13,17,35,30,19,36,18,26)(14,22,41,44,23,34,24,42)(15,16,31,39,27,32,28,33)(20,37,43,29,38,40,25,21);(6,10)(9,11)(14,23)(15,27)(16,32)(22,34)(24,41)(28,31)(33,39)(42,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903913);