Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14)(15,21)(16,28)(17,34)(18,23)(19,35)(20,24)(22,32)(25,39)(26,37)(27,38)(29,36)(30,43)(31,40)(33,42)(41,44);(1,2)(3,4)(6,10)(11,12)(13,15,20,27)(14,21,24,38)(16,29,33,30)(17,18,26,19)(22,39,40,44)(23,37,35,34)(25,31,41,32)(28,36,42,43);(5,7,9,8)(6,11,10,12)(13,16,18,31,20,33,19,32)(14,22,35,42,24,40,23,28)(15,25,17,30,27,41,26,29)(21,36,37,44,38,43,34,39);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,15,20,26,18,27)(14,21,23,34,24,38,35,37)(16,30,32,25,33,29,31,41)(22,36,28,39,40,43,42,44);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,23,24,35)(15,26,27,17)(16,31,33,32)(21,37,38,34)(22,28,40,42)(25,29,41,30)(36,44,43,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fg,e^2=fi,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^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472996);