Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,25)(26,28);(6,9)(10,11)(13,15,16,20)(14,17,19,18)(22,24)(26,27);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,22,25,24)(23,26,28,27);(1,2)(3,4)(5,7)(6,11)(8,12)(9,10)(13,14)(15,17)(16,19)(18,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232465, #72232490, #72232522, … |
| maximal normal subgroup of | #72232465, #72232490, #72232522, … |
| maximal quotient of | #72298180, #72329992, #72330661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1333637);