Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
3782 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)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,26)(23,24,28,27);(8,12)(10,11)(14,17)(15,20)(21,23)(22,24)(25,28)(26,27);(6,7)(10,11)(14,17)(18,19)(22,26)(23,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,18,17,19)(21,23)(22,27)(24,26)(25,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24,25,27)(22,23,26,28)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232525, #72232537, #72232562, … |
| maximal normal subgroup of | #72232525, #72232537, #72232562, … |
| maximal quotient of | #72328941, #72330350, #72333301 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4991503);