Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
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 | |
4591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,13)(11,16,12,15)(17,18)(19,22)(20,24)(21,23);(6,8)(10,13)(11,12)(17,19,21,24)(18,22,23,20);(5,6)(7,8)(9,11,14,12)(10,15,13,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,12,14,11)(10,16,13,15)(17,19)(18,20)(21,24)(22,23);(9,13,14,10)(11,16,12,15)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232107, #72232108, #72232183, … |
| maximal normal subgroup of | #72232107, #72232108, #72232183, … |
| maximal quotient of | #72277972, #72278358, C2×#2032087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1939283);