Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
3846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(10,13)(11,12)(17,18)(19,24)(20,23)(21,22);(1,2,3,4)(6,8)(9,10,14,13)(11,16,12,15)(18,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,13,16)(17,19,21,20)(18,23,22,24);(1,2)(3,4)(9,12,14,11)(10,16,13,15)(17,18)(19,23)(20,24)(21,22);(9,13,14,10)(11,16,12,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,d^2=fgi,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 | #72297166, #72297170, #72297202, … |
| maximal normal subgroup of | #72297166, #72297170, #72297202, … |
| maximal quotient of | #72298528, #72298559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 786452);