Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1921 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,26)(17,22)(18,33)(19,23)(21,28)(24,36)(25,34)(27,41)(29,42)(30,43)(31,38)(32,44)(35,40)(37,39);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,25,31,32)(21,26,36,42)(22,34,38,44)(27,39,30,40)(35,41,37,43);(2,4)(13,16,18,28,19,29,15,24)(14,21,20,26,23,36,33,42)(17,30,32,39,31,27,25,40)(22,37,34,43,38,35,44,41);(5,7,9,8)(6,10,11,12)(13,17)(14,22)(15,25)(16,27)(18,32)(19,31)(20,34)(21,35)(23,38)(24,39)(26,41)(28,40)(29,30)(33,44)(36,37)(42,43);(2,4)(5,8,9,7)(6,10,11,12)(17,31)(22,38)(25,32)(27,30)(34,44)(35,37)(39,40)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594094);