Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3580 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)(6,9)(10,12)(13,14)(15,19)(16,17)(18,20)(22,26)(23,28);(14,17)(18,19)(21,22)(23,24)(25,26)(27,28);(1,2,3,4)(13,15)(14,18)(16,20)(17,19)(21,23)(22,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,20)(17,18)(21,23)(22,24)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,26)(25,27)Polycyclic presentation:
2^9|c^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72300656, #72300669, #72300679 |
| maximal normal subgroup of | #72300656, #72300669, #72300679 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 743402);