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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2185 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)(8,12)(9,10)(13,14,18,19)(15,20,25,34)(16,26,29,21)(17,30,33,22)(23,37,27,35)(24,31,42,36)(28,38,32,39)(40,43,41,44);(2,3)(6,10)(11,12)(13,15,18,25)(14,20,19,34)(16,27,29,23)(17,31,33,36)(21,37,26,35)(22,24,30,42)(28,43,32,44)(38,41,39,40);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,23)(17,32)(18,29)(19,26)(20,35)(22,39)(24,41)(25,27)(28,33)(30,38)(31,43)(34,37)(36,44)(40,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,24)(16,28)(18,33)(19,30)(20,36)(21,38)(23,40)(25,42)(26,39)(27,41)(29,32)(31,34)(35,43)(37,44);(6,10)(11,12)(16,29)(17,33)(21,26)(22,30)(23,27)(24,42)(31,36)(35,37)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468833);