Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C22×C42 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(10,13)(11,15)(18,23)(24,30)(25,26)(31,32);(9,10)(11,14)(12,13)(15,16)(17,18,22,26,21,23,20,25)(19,24,27,31,28,30,29,32);(1,2,3,4)(10,13)(14,16)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,31,30,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239342, #72239348, #72239355, … |
| maximal normal subgroup of | #72239342, #72239348, #72239355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262433);