Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
4010 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,18)(17,20)(21,22,25,27)(23,24,28,26);(5,7,8,12)(6,9,10,11)(13,14,16,18)(15,17,20,19)(21,23,25,28)(22,26,27,24);(6,10)(9,11)(15,20)(17,19)(23,28)(24,26);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,17)(16,20)(18,19)(21,24)(22,23)(25,26)(27,28);(5,7)(6,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232356, #72232368, #72232428, … |
| maximal normal subgroup of | #72232356, #72232368, #72232428, … |
| maximal quotient of | #72323559, #72330265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4373230);