Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
3381 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)(5,6,9,10)(7,11,12,8)(13,14,18,20)(15,16,17,19)(21,22)(23,27)(24,26)(25,28);(5,7,9,12)(6,8,10,11)(13,15)(14,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28);(6,10)(8,11)(14,20)(15,17)(21,22,24,26)(23,25,28,27);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,8)(6,7)(9,11)(10,12)(13,17,18,15)(14,16,20,19)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fg,d^2=fi,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 | #72331957, #72331983, #72332024, … |
| maximal normal subgroup of | #72331957, #72331983, #72332024, … |
| maximal quotient of | #72332692 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990464);