Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
3650 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)(6,8)(7,11)(13,14,17,16)(15,18,20,19)(21,22)(23,27)(24,26)(25,28);(5,6,9,8)(7,10,11,12)(13,15,17,20)(14,18,16,19)(21,23)(22,27)(24,25)(26,28);(7,11)(10,12)(14,16)(18,19)(22,24)(23,28);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(13,16,17,14)(15,19,20,18)(21,24,26,22)(23,25,28,27);(5,8,9,6)(7,10,11,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72328304, #72328358, #72328414, … |
| maximal normal subgroup of | #72328304, #72328358, #72328414, … |
| maximal quotient of | #72329993, #72330098, #72330223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5516381);