Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,12,8,11)(13,14,18,17)(15,19,16,20)(21,22,25,27)(23,28,26,24);(5,7,10,8)(6,11,9,12)(13,14)(15,20)(16,19)(17,18)(21,23)(22,24)(25,26)(27,28);(13,15,18,16)(14,19,17,20)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(5,8,10,7)(6,12,9,11)(13,16,18,15)(14,19,17,20)(21,22)(23,24)(25,27)(26,28);(5,9,10,6)(7,11,8,12)(13,17,18,14)(15,20,16,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=gh,d^2=gi,e^2=fg,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 | #72213086, #72213149, #72267455, … |
| maximal normal subgroup of | #72213086, #72213149, #72267455, … |
| maximal quotient of | #72267414, #72267425, #72267472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1402068);