Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2121 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)(5,6,9,11)(7,12,8,10)(13,14,19,24)(15,26,28,20)(16,21,34,38)(17,25,31,22)(18,23,29,40)(27,41,43,35)(30,44,33,36)(32,39,42,37);(5,6)(7,10)(8,12)(9,11)(13,15,19,28)(14,20,24,26)(16,30,34,33)(17,18,31,29)(21,36,38,44)(22,23,25,40)(27,42,43,32)(35,37,41,39);(1,2,3,4)(13,16,31,42,19,34,17,32)(14,21,22,37,24,38,25,39)(15,27,18,33,28,43,29,30)(20,35,40,44,26,41,23,36);(5,7,9,8)(6,10,11,12)(13,17,19,31)(14,22,24,25)(15,18,28,29)(16,32,34,42)(20,23,26,40)(21,37,38,39)(27,33,43,30)(35,36,41,44);(5,8,9,7)(6,10,11,12)(13,18,31,28,19,29,17,15)(14,23,22,20,24,40,25,26)(16,33,42,43,34,30,32,27)(21,36,37,35,38,44,39,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724176);