Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,21,28,38)(16,29,17,22)(18,34,20,23)(25,43,26,36)(27,39,35,37)(30,44,32,40)(31,41,33,42);(1,2,3,4)(13,15)(14,21)(16,30)(17,32)(18,27)(19,28)(20,35)(22,40)(23,37)(24,38)(25,31)(26,33)(29,44)(34,39)(36,42)(41,43);(6,9)(10,12)(13,16,19,17)(14,22,24,29)(15,25,28,26)(18,31,20,33)(21,36,38,43)(23,41,34,42)(27,32,35,30)(37,40,39,44);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,29)(15,26,28,25)(18,33,20,31)(21,36,38,43)(23,41,34,42)(27,30,35,32)(37,40,39,44);(13,18,19,20)(14,23,24,34)(15,27,28,35)(16,31,17,33)(21,37,38,39)(22,41,29,42)(25,32,26,30)(36,40,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357176);