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 | #45269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 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)(13,14,18,23,19,24,32,40)(15,25,27,37,28,42,29,20)(16,21,31,39,30,38,44,36)(17,22,34,41,33,35,26,43);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,20,24,37)(16,26,30,34)(17,31,33,44)(18,29,32,27)(21,35,38,22)(23,42,40,25)(36,43,39,41);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,30)(14,21,24,38)(15,26,28,34)(17,29,33,27)(18,31,32,44)(20,35,37,22)(23,39,40,36)(25,43,42,41);(5,8,10,9)(6,11,7,12)(13,17,32,26,19,33,18,34)(14,22,40,43,24,35,23,41)(15,16,29,44,28,30,27,31)(20,36,42,38,37,39,25,21);(5,9,10,8)(6,11,7,12)(16,30)(17,33)(21,38)(22,35)(26,34)(31,44)(36,39)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=h,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, 7106344);