Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
5059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,23)(19,20,24,22);(10,14)(12,15)(17,19)(18,22)(20,23)(21,24);(1,2,4,3)(5,6,8,7)(10,14)(11,16)(18,23)(19,24);(5,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23);(1,3,4,2)(5,7,8,6)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232657, #72232693, #72232694, … |
| maximal normal subgroup of | #72232657, #72232693, #72232694, … |
| maximal quotient of | #72330548, #72332990, #72333004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1102405);