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) = 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 | |
3291 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)(6,7)(10,11)(14,18)(16,19)(21,22,26,27)(23,24,25,28);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(22,27)(23,25);(6,7)(10,11)(13,15,17,20)(14,19,18,16)(21,23,26,25)(22,28,27,24);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,16,17,19)(14,20,18,15)(21,25,26,23)(22,28,27,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgh,d^2=gh,e^2=fgh,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 | #72327196, #72327221, #72327272, … |
| maximal normal subgroup of | #72327196, #72327221, #72327272, … |
| maximal quotient of | #72332781 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544663);