Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2331 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)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,32)(20,30,35,26)(25,39,42,44)(28,36,31,38)(33,40,43,41);(5,6)(7,10)(8,9)(11,12)(13,15,17,30,19,27,18,26)(14,20,32,34,23,35,22,24)(16,25,31,40,29,42,28,41)(21,33,36,44,37,43,38,39);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,28)(19,29)(20,33)(22,38)(23,37)(24,39)(26,41)(27,42)(30,40)(32,36)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,30)(16,28,29,31)(20,34,35,24)(21,36,37,38)(25,40,42,41)(33,39,43,44);(7,11)(10,12)(15,27)(16,29)(20,35)(21,37)(24,34)(26,30)(28,31)(36,38)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442875);