Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
3422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,17,19)(15,16,20,18)(21,22,25,24)(23,26,28,27);(5,6)(7,12)(8,9)(10,11)(13,15,17,20)(14,18,19,16)(22,24)(26,27);(6,9)(10,12)(14,19)(15,20)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,15,19,20)(21,24,25,22)(23,27,28,26);(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72278649, #72278836, C2×#4124170, … |
| maximal normal subgroup of | #72278649, #72278836, C2×#4124170, … |
| maximal quotient of | #72278676, #72278787, #72279031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4031366);