Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2164 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)(8,12)(9,10)(13,14,20,26)(15,21,29,22)(16,30,28,38)(17,32,34,42)(18,25,19,24)(23,36,41,35)(27,43,44,40)(31,37,33,39);(1,2)(3,4)(6,10)(11,12)(13,15,20,29)(14,21,26,22)(16,18,28,19)(17,33,34,31)(23,40,41,43)(24,30,25,38)(27,35,44,36)(32,39,42,37);(5,7,9,8)(6,11,10,12)(13,16,19,15,20,28,18,29)(14,22,25,38,26,21,24,30)(17,31,36,27,34,33,35,44)(23,39,42,43,41,37,32,40);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,36,20,34,18,35)(14,23,25,42,26,41,24,32)(15,27,28,33,29,44,16,31)(21,37,30,40,22,39,38,43);(6,10)(11,12)(17,34)(23,41)(27,44)(31,33)(32,42)(35,36)(37,39)(40,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469345);