Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,14,12)(11,16,15,13)(18,20)(22,23);(6,7)(10,12)(11,15)(17,18,21,20)(19,23,24,22);(5,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,22,20,23);(1,2)(3,4)(5,7,8,6)(9,12,14,10)(11,16,15,13);(9,13,14,16)(10,15,12,11)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72297416, #72297442, #72297455, … |
| maximal normal subgroup of | #72297416, #72297442, #72297455, … |
| maximal quotient of | #72298367, #72298378, #72298399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 753786);