Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
4342 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)(7,9)(8,11)(10,12)(13,14,18,17)(15,19,16,20)(22,24)(23,27);(6,7)(8,12)(14,17)(19,20)(23,27)(26,28);(5,6,9,7)(8,10,12,11)(13,15,18,16)(14,19,17,20)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16,18,15)(14,19,17,20)(21,23)(22,26)(24,28)(25,27);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,20,16,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fh,c^2=fgh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232601, #72232609, #72232645, … |
| maximal normal subgroup of | #72232601, #72232609, #72232645, … |
| maximal quotient of | #72328841, #72329004, #72329988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2721056);