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 |
| prime radical | C2×#3321 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1833 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)(7,10)(8,11)(9,12)(13,14,18,23,19,22,33,40)(15,24,27,36,26,41,17,20)(16,28,32,44,31,21,37,39)(25,43,30,35,29,42,34,38);(5,6)(7,10)(8,11)(9,12)(13,15,19,26)(14,20,22,36)(16,29,31,25)(17,33,27,18)(21,38,28,35)(23,41,40,24)(30,37,34,32)(39,42,44,43);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16)(14,21)(15,25)(17,34)(18,32)(19,31)(20,35)(22,28)(23,39)(24,42)(26,29)(27,30)(33,37)(36,38)(40,44)(41,43);(5,8,10,9)(6,11,7,12)(13,17,18,15,19,27,33,26)(14,20,23,24,22,36,40,41)(16,30,32,29,31,34,37,25)(21,38,39,43,28,35,44,42);(5,9,10,8)(6,11,7,12)(14,22)(15,26)(16,31)(17,27)(23,40)(32,37)(35,38)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,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, 7106345);