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) = 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 | |
3554 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)(5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,26)(24,28)(25,27);(1,2)(3,4)(6,7)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,25,28)(22,26,27,24);(1,2,3,4)(5,6,9,7)(8,11,12,10)(13,16,18,17)(14,15,19,20)(21,24)(22,23)(25,26)(27,28);(5,7,9,6)(8,11,12,10)(13,17,18,16)(14,15,19,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=f,d^2=ghi,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 | #72232640, #72232652, #72232658, … |
| maximal normal subgroup of | #72232640, #72232652, #72232658, … |
| maximal quotient of | #72329938, #72332854 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2284704);