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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3712 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)(5,6,10,9)(7,8,12,11)(13,14)(15,19)(16,17)(18,20)(21,22,25,27)(23,28,24,26);(5,7,10,12)(6,11,9,8)(14,17)(18,19)(21,23)(22,26)(24,25)(27,28);(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,20)(17,18)(21,22)(23,26)(24,28)(25,27);(5,9,10,6)(7,11,12,8)(13,14)(15,18)(16,17)(19,20)(21,24,25,23)(22,26,27,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,e^2=fgh,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 | #72232479, #72232501, #72232531, … |
| maximal normal subgroup of | #72232479, #72232501, #72232531, … |
| maximal quotient of | #72329054, #72330691 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 690733);