Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,27)(17,19)(18,31)(20,24)(21,28)(22,30)(25,37)(26,36)(29,38)(32,34)(33,35)(39,42)(40,44)(41,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(15,24)(16,28)(18,30)(20,36)(21,38)(23,34)(26,32)(27,35)(29,33)(37,43)(40,42);(13,15)(14,20)(16,25)(17,26)(18,32)(19,34)(21,37)(22,36)(23,31)(24,30)(27,39)(28,40)(29,41)(33,42)(35,43)(38,44);(2,4)(13,16,17,29)(14,21,19,35)(15,25,26,41)(18,33,30,28)(20,37,34,43)(22,38,31,27)(23,39,36,44)(24,40,32,42);(6,10)(9,11)(14,19)(15,26)(21,35)(22,31)(24,32)(25,41)(27,38)(40,42)Polycyclic presentation:
2^9|b^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421634);