Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(5,6,8,7)(9,10,15,20)(11,21,25,30)(12,26,29,33)(13,16,14,19)(17,24,31,22)(18,28,34,27)(23,37,39,32)(35,38,40,36);(1,2,4,3)(5,7,8,6)(10,16)(11,22)(13,14)(17,31)(18,26)(19,20)(23,36)(24,25)(27,28)(32,37)(33,34)(38,39);(6,7)(9,11,15,25)(10,17,20,31)(12,23,29,39)(13,22,14,24)(16,30,19,21)(18,32,34,37)(26,40,33,35)(27,38,28,36);(1,3,4,2)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,26)(17,32)(19,33)(20,34)(21,35)(22,36)(24,38)(25,39)(30,40)(31,37)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=efhi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285309);