Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,26,30,21)(16,23,17,22)(18,34,20,24)(27,36,32,42)(28,37,44,43)(29,38,35,39)(31,40,33,41);(1,2,4,3)(7,12)(9,11)(13,15)(14,21)(16,27)(17,32)(18,29)(19,30)(20,35)(22,36)(23,42)(24,38)(25,26)(28,33)(31,44)(34,39)(37,40)(41,43);(5,7,8,12)(6,9,10,11)(13,16,19,17)(14,22,25,23)(15,27,30,32)(18,33,20,31)(21,36,26,42)(24,41,34,40)(28,35,44,29)(37,38,43,39);(6,10)(9,11)(13,17,19,16)(14,23,25,22)(15,28,30,44)(18,33,20,31)(21,37,26,43)(24,41,34,40)(27,35,32,29)(36,38,42,39);(1,3,4,2)(13,18,19,20)(14,24,25,34)(15,29,30,35)(16,31,17,33)(21,38,26,39)(22,40,23,41)(27,44,32,28)(36,37,42,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7441073);