Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
4235 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)(7,10)(8,12)(9,11)(14,18)(15,16)(21,22,24,26)(23,27,28,25);(5,7,9,12)(6,10,11,8)(13,14)(15,19)(16,20)(17,18)(23,28)(25,27);(5,7)(6,8)(9,12)(10,11)(13,15,17,16)(14,19,18,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,15)(14,20,18,19)(21,23)(22,25)(24,28)(26,27);(5,8)(6,7)(9,10)(11,12)(13,14)(15,20)(16,19)(17,18)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234814, #72234818, #72234828, … |
| maximal normal subgroup of | #72234814, #72234818, #72234828, … |
| maximal quotient of | #72328594, #72328631, #72330036 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243030);