Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,14,12)(11,16,13,15)(17,18)(19,20)(21,24)(22,23);(1,2,3,4)(6,8)(9,11)(10,15)(12,16)(13,14)(17,19,23,24)(18,21,22,20);(5,6)(7,8)(9,10,14,12)(11,16,13,15)(17,20,23,21)(18,24,22,19);(1,2,3,4)(9,12,14,10)(11,16,13,15)(17,21,23,20)(18,24,22,19);(1,2)(3,4)(9,13,14,11)(10,15,12,16)(17,22,23,18)(19,21,24,20)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=fi,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72278969, #72297349, #72297388, … |
| maximal normal subgroup of | #72278969, #72297349, #72297388, … |
| maximal quotient of | #72279181, #72298239, #72299080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2959846);