Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3989 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)(6,9)(10,11)(14,17)(15,19)(22,26)(24,27);(5,6,8,9)(7,11,12,10)(15,19)(18,20)(22,26)(23,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,26)(23,27,28,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|b^2=f,c^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232281, #72232283, #72232284, … |
| maximal normal subgroup of | #72232281, #72232283, #72232284, … |
| maximal quotient of | #72328356, #72329833, #72329838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1945955);