Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
2327 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,33,39)(15,25,27,37,28,41,29,20)(16,21,31,36,30,38,42,44)(17,32,35,43,34,22,26,40);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,24,37)(16,26,30,35)(17,31,34,42)(18,29,33,27)(21,22,38,32)(23,41,39,25)(36,43,44,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,30)(14,21,24,38)(15,26,28,35)(17,29,34,27)(18,31,33,42)(20,22,37,32)(23,36,39,44)(25,40,41,43);(7,12)(9,11)(13,17,33,26,19,34,18,35)(14,22,39,43,24,32,23,40)(15,16,29,42,28,30,27,31)(20,36,41,21,37,44,25,38);(6,10)(9,11)(16,30)(17,34)(21,38)(22,32)(26,35)(31,42)(36,44)(40,43)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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903977);