Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ 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:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,23)(16,21)(17,30)(18,20)(19,26)(22,31)(24,38)(25,37)(27,42)(28,34)(29,44)(32,41)(33,39)(35,43)(36,40);(5,7)(6,9)(8,11)(10,12)(13,15,17,26,18,24,31,39)(14,19,22,23,20,33,30,38)(16,27,29,43,28,25,40,41)(21,35,36,42,34,32,44,37);(1,2,3,4)(7,11)(9,12)(14,20)(15,24)(21,34)(22,30)(25,27)(26,39)(36,44)(41,43);(7,11)(9,12)(13,16)(14,21)(15,25)(17,29)(18,28)(19,32)(20,34)(22,36)(23,37)(24,27)(26,41)(30,44)(31,40)(33,35)(38,42)(39,43);(16,28)(21,34)(25,27)(29,40)(32,35)(36,44)(37,42)(41,43)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364489);