Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
3646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(14,16)(15,19)(21,22)(23,25)(24,26)(27,28);(2,3)(6,9)(10,11)(13,14,17,16)(15,18,19,20)(21,23,26,28)(22,27,24,25);(1,2,4,3)(6,9)(10,11)(15,19)(18,20)(22,24)(23,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,22)(23,27,28,25);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,20,19,18)(21,25,26,27)(22,28,24,23)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fi,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297737, #72297742, #72297766, … |
| maximal normal subgroup of | #72297737, #72297742, #72297766, … |
| maximal quotient of | #72298329 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 837867);