Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,18,23,19,24,17,22)(15,25,29,37,30,36,28,20)(16,21,31,38,32,39,33,40)(26,41,43,35,27,42,44,34);(5,6)(7,11)(8,10)(9,12)(13,15,19,30)(14,20,24,37)(16,27,32,26)(17,29,18,28)(21,35,39,34)(22,25,23,36)(31,43,33,44)(38,41,40,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,21,24,39)(15,26,30,27)(17,33,18,31)(20,34,37,35)(22,40,23,38)(25,41,36,42)(28,44,29,43);(7,12)(9,11)(13,16)(14,21)(15,27)(17,33)(18,31)(19,32)(20,35)(22,40)(23,38)(24,39)(25,42)(26,30)(28,43)(29,44)(34,37)(36,41);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,28,30,29)(16,31,32,33)(20,36,37,25)(21,38,39,40)(26,43,27,44)(34,41,35,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407409);