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 | |
3466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14,16,18)(15,19,20,17)(21,22)(23,24)(25,26)(27,28);(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(22,26)(23,28);(6,7)(10,11)(15,20)(17,19)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,17,18,19)(21,24)(22,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=fg,c^2=h,d^2=fg,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 | #72256739, #72256751, #72256791 |
| maximal normal subgroup of | #72256739, #72256751, #72256791 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517324);