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 | |
4116 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)(7,11)(8,12)(9,10)(13,14,17,20)(15,19,16,18)(21,22,24,25)(23,27,28,26);(6,10)(11,12)(22,25)(26,27);(5,7,9,8)(6,11,10,12)(13,15,17,16)(14,18,20,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,9,7)(6,11,10,12)(13,16,17,15)(14,19,20,18)(21,23,24,28)(22,26,25,27);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,19)(17,20)(21,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^9|a^2=g,c^2=fg,d^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 | #72232520, #72232522, #72232566, … |
| maximal normal subgroup of | #72232520, #72232522, #72232566, … |
| maximal quotient of | #72298180, #72330044, #72330305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1329501);