Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
3568 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)(8,10)(9,12)(13,14,17,19)(15,16,20,18)(21,22,25,28)(23,27,24,26);(5,7,10,12)(6,9,8,11)(13,15,17,20)(14,18,19,16)(23,24)(26,27);(6,8)(7,12)(15,20)(16,18)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,16)(14,15)(17,18)(19,20)(21,24,25,23)(22,27,28,26);(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256745, #72256758, #72256778, … |
| maximal normal subgroup of | #72256745, #72256758, #72256778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4374223);