Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,17,19,24,18,23)(15,25,36,21,28,40,27,38)(16,29,32,31,34,22,33,35)(26,42,43,39,30,41,44,37);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,24,38)(16,30,34,26)(17,25,23,40)(18,36,20,27)(22,39,29,37)(31,41,35,42)(32,43,33,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,34)(14,22,24,29)(15,26,28,30)(17,35,23,31)(18,33,20,32)(21,37,38,39)(25,41,40,42)(27,44,36,43);(7,12)(9,11)(13,17,20,24,19,23,18,14)(15,21,36,40,28,38,27,25)(16,31,32,22,34,35,33,29)(26,39,43,41,30,37,44,42);(6,10)(9,11)(13,18,19,20)(14,23,24,17)(15,27,28,36)(16,32,34,33)(21,25,38,40)(22,35,29,31)(26,43,30,44)(37,42,39,41)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972342);