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×#3540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3088 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)(13,14,18,23,19,24,16,21)(15,25,26,35,28,38,33,20)(17,30,32,36,31,22,29,37)(27,41,40,34,42,39,44,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,35)(16,26,18,33)(17,27,31,42)(21,25,23,38)(22,34,30,43)(29,40,32,44)(36,39,37,41);(2,4)(6,9)(7,11)(13,16,19,18)(14,21,24,23)(15,26,28,33)(17,29,31,32)(20,25,35,38)(22,36,30,37)(27,40,42,44)(34,39,43,41);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,27)(16,29)(18,32)(19,31)(20,34)(21,36)(23,37)(24,30)(25,39)(26,40)(28,42)(33,44)(35,43)(38,41);(6,9)(10,12)(17,31)(22,30)(27,42)(29,32)(34,43)(36,37)(39,41)(40,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420564);