Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,9)(7,11,12,10)(14,17)(15,19)(22,25)(26,28);(6,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(6,9)(10,11)(14,17)(18,20)(23,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=gi,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232292, #72232308, #72236741, … |
| maximal normal subgroup of | #72232292, #72232308, #72236741, … |
| maximal quotient of | #72328712, #72331057, C2×#582589 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489785);