Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
3455 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)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,25)(24,26)(27,28);(5,7,9,12)(6,10,8,11)(13,14,16,17)(15,19,20,18)(23,27)(25,28);(6,8)(10,11)(14,17)(18,19)(21,22,24,26)(23,25,27,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15,16,20)(14,18,17,19)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28)Polycyclic presentation:
2^9|b^2=f,c^2=fg,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72300636, #72300645, #72300659 |
| maximal normal subgroup of | #72300636, #72300645, #72300659 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188593);