Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3066 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)(7,12)(8,11)(9,10)(13,14)(15,17)(16,20)(18,19)(22,26)(24,28);(5,7)(6,8)(9,12)(10,11)(14,19)(15,16)(22,26)(23,27);(6,9)(7,11)(13,15,18,16)(14,17,19,20)(21,22)(23,24)(25,26)(27,28);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16,18,15)(14,17,19,20)(21,23)(22,24)(25,27)(26,28);(5,9,10,6)(7,12,11,8)(13,17)(14,15)(16,19)(18,20)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|c^2=gh,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72333852, #72333913, #72333918 |
| maximal normal subgroup of | #72333852, #72333913, #72333918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2234097);