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 | |
3244 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)(5,6,8,10)(7,11,12,9)(14,17)(18,20)(21,22,26,25)(23,28,24,27);(5,6)(7,9)(8,10)(11,12)(15,19)(18,20)(21,22)(23,28)(24,27)(25,26);(7,12)(9,11)(13,14,16,17)(15,18,19,20)(21,23,26,24)(22,27,25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24,26,23)(22,27,25,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,28,24,27)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=h,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 | #72333992, #72334004, #72334039 |
| maximal normal subgroup of | #72333992, #72334004, #72334039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5537636);