Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26,21,25,32,27)(19,20,29,31,28,23,24,30);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,28)(18,23,25,20)(22,29,32,24)(26,30,27,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,16,14)(17,20,21,23)(18,24,25,29)(19,27,28,26)(22,31,32,30);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(18,25)(20,23)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268730);