Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213818 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,19,22,15)(12,23,26,37)(13,17)(14,18)(16,25,31,40)(20,35)(21,32,27,30)(24,38)(28,33)(29,36)(34,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,24)(13,27)(14,22)(16,29)(17,32)(18,19)(20,26)(21,28)(23,35)(25,36)(30,33)(31,34)(37,38)(39,40);(1,4,5,6)(2,3,7,8)(9,12)(10,16,18,31)(11,20)(13,25)(14,26)(15,29,19,34)(17,23,33,37)(21,36)(22,24)(27,39)(28,40)(30,38,32,35);(2,7)(3,8)(10,17,18,33)(15,30,19,32)(16,23,31,37)(29,38,34,35)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294640);