Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×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 | |
3619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,11,12)(13,14,18,17)(15,19,16,20)(22,25)(23,24);(6,8)(10,12)(14,17)(19,20)(21,22)(23,28)(24,27)(25,26);(1,2,4,3)(13,15,18,16)(14,19,17,20)(21,23,26,24)(22,27,25,28);(1,2,4,3)(5,7,9,11)(6,10,8,12)(13,16,18,15)(14,19,17,20)(21,24,26,23)(22,27,25,28);(1,3,4,2)(5,8,9,6)(7,10,11,12)(13,17,18,14)(15,20,16,19)(21,25,26,22)(23,28,24,27)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^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 | #72328703, #72328776, #72328798, … |
| maximal normal subgroup of | #72328703, #72328776, #72328798, … |
| maximal quotient of | #72330189 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1383447);