Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2920 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,15)(12,14)(13,16)(17,20)(18,23)(19,21)(22,24);(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(1,3,4,8,5,2,6,7)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(2,3)(7,8);(10,16)(11,18)(14,17)(19,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=eh,f^a=fh,f^b=fh,d^c=di,e^c=eh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72216715, #72216884, #72216937, … |
| maximal normal subgroup of | #72216715, #72216884, #72216937, … |
| maximal quotient of | #72318429, C2×#6345071, #72319418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252267);